Adaptive Fission: Post-training Encoding for Low-latency Spike Neural Networks
Authors: Yizhou Jiang
Feng Chen
Yihan Li
Yuqian Liu
Haichuan Gao
Tianren Zhang
Ying Fang
Venue: NeurIPS 2025 poster ()
Keywords: spiking neural network
ann-to-snn conversion
Primary area: neuroscience_and_cognitive_science
TL;DR: This paper introduces a post-training encoding method that reduces the number of time-steps required for spiking neural networks.

Abstract: Spiking Neural Networks (SNNs) often rely on rate coding, where high-precision inference depends on long time-steps, leading to significant latency and energy cost—especially for ANN-to-SNN conversions. To address this, we propose Adaptive Fission, a post-training encoding technique that selectively splits high-sensitivity neurons into groups with varying scales and weights. This enables neuron-specific, on-demand precision and threshold allocation while introducing minimal spatial overhead. As a generalized form of population coding, it seamlessly applies to a wide range of pretrained SNN architectures without requiring additional training or fine-tuning. Experiments on neuromorphic hardware demonstrate up to 80\% reductions in latency and power consumption without degrading accuracy.

Adaptive Fission: Post-training Encoding for Low-latency Spike Neural Networks
Yizhou Jiang'
Feng Chen
Yihan Li
Yugian Liu'
Haichuan Gao
Tianren Zhang' Ying Fang**
Department Automation, Tsinghua University, Beijing, China
Beijing QianJue Technology Co., Ltd.
College of Computer and Cyber Security, Fujian Normal University, Fuzhou, China (jiangyz20, 1yh19, Liuyuqian21)mmaiii amaals.......m...mmmmmammmaamlls tsinghua. edu. en haichuan. gao@qj-robote com, chenfeng@mail tsinghua. edu. cn trzhang@mail tsinghua. edu.cn, fy200f jnu. edu. cn
Abstract
Spiking Neural Networks (SNNs) often rely rate coding, where high-precision inference depends on long time-steps, leading to significant latency and energy cost-espeyyllly for ANN-to-SNN conversions. To address this, propose Adaptive Fission, post-training encoding technique that selectively splits high- sensitivity neurons groups with varying scales and weights. This enables neuron-specific, on-demand precision and threshold allocation while introducing minimal spatial overhead. As a generalized form of population coding, seamlessly applies to wide range of pretrained SNN architectures without requiring additional training or fine-tuning. Experiments on neuromorphic hardware demonstrate up 80% reductions in latency and power consumption without degrading accuracy.
Introduction
Unlike traditional artificial neural networks (ANNs) that use seating--pint activations, spiking neural networks (SNNs) encode activations as binary spike sequences over time, promising energy-efficiee intelligence on neuromorphic hardware [3 5]. However, they still lag behind ANNs in precision and performance due to inefficient rate-based encoding. While T-bit quantized ANN represents 2n distinct values, an SNN needs T time-steps with 1-bit outputs per step to encode only T+1 discrete states, illustrated in Fig. [ This makes high-precision inference slow and energy-intensive complex tasks such as generation
20
recent efforts aim train SNNs from seratch thereby reducing reliance on temporal precision. However, in many real-world cases such as open- vocabulary detection, SNNs converted from large, pretrained ANNs (e.g., applications CLIP [35 24 (00 48)). where retraining is infeasible due to proprietary data computational cost. To address this, existing post-training methods either modify spike bits 42 41 or increase firing steps locally (27], but yield only modest gains. Moreover, most neuromorphic chips lack support for multi- bit, multi-thresholl neurons or dynamic time-stepy steps. Therefore, a hardware- friendly post- bost-training solution for low-latency deployment remains critically needed.
Inspired by training-aware quantization
9 51
with variable bit-lengths or time-steps
[45
37
"Corresponding authors.
'Code available http::///gghhto.oco///////aa/aap
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Quantiiiito Encoding:
Adaptive Fission Encoding:
weight
2
Adaptive weight
20
2.8
2.75
2
1
bit length
0.05
0.6
Rate Encoding:
2.8 -
+ 2.8
Adaptive bit length
0.2
2.8
bit length
Figure 1: An overview of encoding methods. Adaptive Fission combines the sparsity of SNN rate encoding with the efficiency of ANN quantization, enabling variable bit-lengths and weights.
As a classical yet underutilized alternative, population coding distributes representation across groups of neurons, replacing temporal integration with spatial parallelism and thus reducing latency. Prior works 40 30 24 typically adopt static group structures with fixed scales, weights, and thresholds, resulting in substantial memory overhead and hindering practical deployment. In contrast, we propose Adaptive Fission, a flexible, generalized population-codiie splits high-sensitivity neurons into dynamically sized groups and allocates precision and thresholds on demand. This neuron-wise bit allocation enables fexed-precision inference while remaining compatible with binary neuron models. Our analysis shows it approaches the theoretical capacity of population coding with only moderate spatial overhead.
strategy that selectively
To implement Adaptive Fission, we design a two-stage iterative pipeline to address two key challenges: identifying which neurons should undergo fission and determining the scales and thresholds within each neuron group. In the first stage, estimate and rank neuron sensitivity by jointly analyzing gradients and residual membrane potentials, prioritizing those with the greatest impact. This targeted reallocation achieves capacity comparable to fixed- -enghhh population encoding with only 20% addi- tional neurons. In the second stage, we derive an optimality condition for assigning thresholds to the newly created sub-neurons, based the cumulative distribution of membrane potentials. Although analytically intractable, we approximate the solution using Monte Carlo sampling combined with Newton's 's method, yielding minimal encoding error.
By enhancing representational precision under strict latency constraints, Adaptive Fission enables faster, more energy-efficient inference. As post-training technique, it is compatible with both ANN-to-SNN conversions and directly trained SNNs, and can be combined with quantization and pruning to offset the modest increase in neuron count. Deployments chip 34 demonstrate up to 80% reductions in latency and energy consumption without compromising accuracy. Beyond classification, Adaptive Fission also supports generative tasks like image synthesis, highlighting its potential for high-throughput neuromorphic computing.
Lynxi HP201 neuromorphic
Our key contributions are follows:
1. We propose Adaptive Fission, a post-training population-coding method that assigns variable bit lengths and weights, with theoretical guarantees of exponential error reduction.
2. We introduce practical, hardware-aware encoding pipeline for parallel SNN acceleration, compatible with both ANN-to-SMN -----NNN conversion and directly trained SNNs.
3. Evaluation on neuromorphic platform demonstrates up to 80% reductions latency and power consumption while maintaining competitive accuracy.
2 Related Work
2.1 Spike Encoding
Rate coding 1 remains the dominant strategy in SNNs, where all spikes contribute equally with temporal information discarded. Despite simplicity and robustness, it suffers from low information density. Variants such signed or ternary spikes 42 [4] and variable spike amplitudes

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enhance expressivity but often compromise hardware compatibility. Alternatvely, population coding leverages multiple neurons to jointly represent scalar values. While promising, existing works typically employ fixed-scale ensembles 45 37 or rely hand-crafted heuristics for neuron substitution (24), limiting precision scalability. Besides, temporal coding [12 uses spike timing rather than frequency to improve encoding density, but it currently underperforms rate coding. Our method can be viewed as a generalized framework of population coding with adaptive, hardware-aware support.
or thresholds [1] 47 8
39 32 22
2.2 Latency Reduction in SNNs
Reducing the number of simulation steps is critical for latency and energy-efficieney. Existing efforts can be categorized into three main types: (1) Post-training conversions introduce burst spikes 29 or negative spikes 422. but still typically require 32 time-steps to match ANN- level accuracy. (2) Quantization-wware conversions emulate low-bit activations via modified activation functions [77 4, or employ pre-charged neurons [23] to reduce inter-layer delays,
bringing time-steps down to 8-16 but necessitating training from scratch. (3) Direct training
approaches treat SNNs as recurrent networks and use surrogate gradients [33 52 or synaptic
plasticity 2). with recent work incorporating variable-bit activations 44 37.. Despite achieving
2-8 time-steps, they suffer from training instability and scalability issues in larger networks [46]. Our method complements these approaches by offering post-training optimization without retraining architectural modifications, and is fully compatible with both converted and directly trained SNNs.
Preliminaries
As our method operates at the neuron level, we first formulate the model from the perspective of individual neurons.
ANN models: In standard feedforward ANN with ReLU activation, each neuron receives input from presynaptic activations dyre via synaptic weights w The total input Jonn and activation a are:
Jonn =
Σ a pret
a = max(qann: 0}.
SNN models with weighted spikes: We focus on soft-reset Integrate-and-FF (IF) neurons 36 16
and adopt the weighted-spike formulation commonly used in ANN-SNI SNN conversion (28). Let 0 be the firing threshold, and (t) the presynaptic spike at time-step t. The total input is qann (f)
E, w s spre (t), and membrane dynamics are defined as:
Charging: Utemp (t) u(t 1)-1 + (e)n(t)).
(2)
Firing: s(t) 0 0 H (Vtempp( (t) 8),
(3)
v() Utemp (1) -(():
where vtt is the membrane potential after step f,, and H.)) the Heaviside function. Here s(t) is not binary {0, 1} but weighted (0, f). To support binary-only hardware, we absorb 0 into synaptic weights:
w = Ow, s(t) = s(t)/0 = H (t)emp()) -)) E 10.
(5)
For simplicity, we continue using the f0, 0} notation, where 0 also denotes the effective output weight.
SNN and Quantized ANNs: In rate coding, a neuron's activation intensity is represented by its average firing rate over T time-steps, i.e., s ΣX s(t) Assuming E. qonn (1) Gann a, the
SNN activation approximates a 7-level quantized ANN:
((T) 0 clip
(6)
T
T
Thus, the firing rate approximates the quantized ANN activation, with quantization error stemming
from and reflected in the residual potential v()). Increasing T distributes activation across more steps, allowing smaller threshold 0 x max(a)/T for finer quantization.
3

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Population Encoding Models
Most SNNs assign a single binary neuron to each activation, limiting precision under temporal constraints. We propose a generalized population coding framework where multiple binary neurons jointly encode a single activation, enabling finer quantization while preserving hardware-fries friendll binarization. Unlike prior methods that require retraining weight updates, our approach is entirely post-training and model-agnostic,, making it applicable to both converted and directly trained SNNs.
4.1 Static Models and Error Analysis
To understand the precision limits of multi-neuron representations, we first isolate static encoding behavior, independent of spike timing.
Definition 4.1 (Encoding Error). Consider group of k neurons with distinct thresholds 0; R+ Let ni denote the spike count of neuron i within 7 time-steps. The set of representable values is:
Oan: + E {0, 1........ T) {.........T))>
s =
0.
where b is constant bias. Given input distribution P(g) with O, the expected encoding error is:
Byg
min (q s)) $ES,eSq
(8)
In this context, a conventional IF neuron is simplified case with A: 1. For fixed 0, the optimal firing count is n clip(L T). Assuming uniform feature distribution, the expected encoding error can be directly computed, aligning with previous results 77].
Proposition 4.2 (Optimal Error of Single Neuron). If (O, (max), the minimal error of a single neuron is given by +TT+1)
Smaze with the threshold and bias 0 = 2r and b = T.
Extending to multi neltin encoding (Def. 4.1 the absence of fixed firing patterns across neurons prevents direct computation individual spike counts n,, Yct under the uniform assumption, an optimal configuration can still be derived without explicit firing rules.
Proposition 4.3 (Optimal Error of Neuron Groups). For k-neuron group encoding UO, (max), the optimal error r, threshold 0g and bias b are given by: 9max (T + 1)5" b=
qmax
2(T + 1) 0,
As the number of neurons increases, the eтror decreases exponentially, yielding far substantial precision gains compared to the linear improvement from increasing time-steps T. This result, however, captures only aggregate encoding after all 7 steps, providing optimal hindsight solution while ignoring spike dynamics.
4.2 Temporal Dynamic Encoding Models
Previous encoding analyses typically assume full knowledge of the input g over all time-steps. In contrast, practical SNN inference operates under an online regime, where input incrementally accumulated and spikes are emitted as soon as the membrane potential crosses the threshold, This scenario can be viewed
special case of Proposition 4.3 with T 1.
Proposition 4.4 (Optimal Error with Temporal Dynamics), For k-neuron group, assume the residual membrane potential at the final hellows P (Vteep((T) (T)) E UO, Tmax r, threshold Og and bias are given by
). The optimal error
Tmax
2max b T 2
T 2 0
(10)
This proposition implies that exponentially increasing precision (2 levels) can still achieved under online temporal encoding by assigning thresholds powers of two. Moreover, it provides

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k((10))
Casrade Model
Parallel Modd
Indicator: (42)
Potential
81
Potential
Indicator
v600
((t)
(x)
((t)
sy(c)
(2)
(t)
(v (+)
% Corget
A.+.
(t)
S2(t)
(+)
D2(v (+))
Model
Response
(c)
(x(e)
(4
(x)
0,+iy
Figure 2: Dynamies of our proposed neuron groups. Neurons are arranged in a hierarchical structure based on descending thresholds, and can be parallelized by pre-compiled indicators. Their response of input spikes forms a multi-level step function as an uneven quantization.
theoretical support for recent neuron neuron-grouping grouping heuristies [30]. However, these results critically rely on the assumption of uniformly distributed residual potential, which rarely holds in practice, where long-tailed or multi-modal distributions are more common.
To generalize the analysis to arbitrary input distributions, a more explicit mathematical formulation of neuron group's 's temporal dynamics is necessary. While multiple formulations possible, adopt a descending-threshol population model (Fig. (2). where the residual input is successively propagated neurons with lower thresholds after higher-threshold neurons fire. Although this formulation is inherently sequential, it admits an equivalent parallel implementation that preserves efficiency.
Definition 4.5 (Spiking Neuron Group Model). Consider group of k neurons with descending thresholds 0 2 82 2 ... 2 0z- At time-step f, let g(t) denote the total synaptic input, (t) the residual potential of neuron and (t) its weighted output spike.
Initialize the first neuron's potential as 0n (1) = (1). Then, for all ( = I...., andd i
Firing:
st( = 0; H (v:(t) - 0:),
(11)
Transfer:
(1) = (1) - ((),,,
(12)
Charging: (1 + 1) = vx)(() (4) a(l + 1).
(13)
The total output stt) is the sum of individual weighted spikes:
s(t) Σ st()).
(14)
Although the above dynamics is presented as sequential mascade-where higher-threshon neurons fire first and residual potential propagated downward-the computation can equivalently be realized in fully parallel manner. Specifically, each neuron's 's firing condition depends solely on the total input and the predefined threshold set, eliminating the need for explicit synchronization with upstream firing events. This property enables the cascade to be "precompiled" into lightweight parallel comparison logic. Formally, once the thresholds O, are fixed, all spike firings can expressed as binary indicator functions II composed of Heaviside terms:
f-1 H( - - -m,0,)-((ur()) j=1
1-1 M,, - -.)) (15)
sc(1) = 02 00- (() = -0,- Σ 10.
3=1
This representation requires only comparisons against fixed thresholds and thus introduces negligible overhead on neuromorphic hardware. The resulting hierarchical model forms the foundation for adaptive neuron encoding. By tuning the number of neurons and the threshold set 0 for each activation, precision can be flexibly allocated without retraining or modifying weight values. In the following section, we demonstrate how determine the optimal configuration of this structure for each neuron based on post-training statistics.
Methodology: Adaptive Fission Encoding
5.1 Overview
In practical networks, individual neurons exhibit diverse activation distributions and precision re- quirements, making it intractable to derive optimal bit-lengths or threshold assignments analytically.
5

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Stage 1: Sensitivity Estimation
Stage 2: Threshold Fission
1. Before
2. After
1. SNNForward
Resichall polential
AfterT stpp
PyCo
200
Py00
Impatt
Seat
()va(T)
PX
Pa(O
2. ANN Backward
Output
Output
Grea
Gradients
d dfea
Inpott
Beal
Erur
9MM
Impor
0g
Irput
Figure 3: Overview of the iterative stages in Fission Encoding pipeline. Sensitivity Estimation identifies neurons with high impact using residual potentials and gradients. Threshold Fission splits selected neurons and optimizes their thresholds based on their activation distribution. The process progressively enhances precision while preserving binary spike constraints.
To address this, adopt two-stage iterative pipeline, termed Adaptive Fission, illustrated in 3 Analogous to post-training calibration in ANN quantization [20] the first stage employs a small calibration set estimate each neuron's 's precision demand based its contribution to the output error. This calibration procedure is entirely training-free free, as it involves no weight updates and only collects activation statistics, thereby requiring a small number of samples and incurring negligible computational overhead. In the second stage, we selectively allocate additional neurons and adjust their thresholds minimize the expected residual potential. The algorithm progressively splits the most influential neurons into two units with reduced weights in each iteration. This neuron "fission" mechanism achieves principled balance between precision and resource consumption while maintaining the original network's 's accuracy.
Stage Sensitivity Estimation and Selection
To prevent excessive scale increase, we consider the inherent redundancy in deep networks, and focus precision enhancement on neurons with the most significant impact on output accuracy, introducing "sensitivity" metric to quantify such impact. Specifically, its estimation involves passes. In the first, a standard SNN inference records the residual potential (eea(T) after steps. In the second pass, temporarily substitute spiking neurons with ReLUs to enable ANN-like gradient backpropagation. Denoting the continuous activation of neuron as Ofea. the sensitivity diea can be approximated first-order Taylor expansion:
propagation
Bl Aout (afea) aout (afea u(T))ll2 El| doout dafea
fea
Ufea Uffaa(T) (2)
(16)
where the expectation is taken over calibration samples. Intuitively, this measures how much the residual potential at each activation neuron affects the final output. Neurons are ranked by a feas and only top fraction (fission rate) are selected for further enhancement.
5.3
Stage 2: Optimal Threshold Fission
For each neuron group selected for fission, we add one extra neuron to redistribute thresholds and achieve finer-grained representation. Unlike prior analysis, we estimate the potential distribution directly from calibration data. Because residual potential is mainly determined by the smallest threshold, fission is applied to the last neuron in the group with threshold fotd- We split it into two with thresholds 9₁ > 0x+1 such that 0₂ + 8x+1 = folds preserving total contribution. Given the residual potential uk (T). the improved quantization reduces residual error as:
Au (0g 0 H(v 0x) 0x+ (v (x+1).
(17)
and the objective is therefore determine 02 that maximizes the expected error reduction, E[Au].
Since Ar non-differentiable, closed-form solutions are infeasible. Instead, we derive the optimal thresholds using the cumulative distribution of potential.

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Theorem 5.1 (Optimal Fission Threshold). Let F(.) denote the cumulative distribution function of residual potential v. The optimal threshold 0g given by solution of equation:
2F(0,) - F(ootd 0x) - -1= 0,
(18)
and the maximum expected reduction in residual potential is:
E[Au] = Oatd - F(o)),
(19)
which is bounded by:
Av E [
(20)
Remark 5.2. Detailed proof in the Appendix A..3 an integral over the range of v, then transforming into the cumulative distribution function F(v):
We first substituting the expectation of Ae with
Ootd
Jr 0x+1p(v)do
Bold
E[Au]
(0x 0 0x+1)p(u)de +
6x
JOLI
0 [1 F(O+)] (fold - 0k) (F((s))()))))))))))) F(Boid 0x)]-
The extremum of 0g = arg maxo, El [ can be obtained by differentiation. As F(v) is estimated from discrete samples, its derivative p(v) can be neglected, yielding:
dB de
(2) - F(foia - 0x) - = 0,
(21)
which provides the optimal solutions in Eq 18
This result implies that each additional neuron introduced by fission encoding can reduce the quan- tization error by at least half, even under the worst-case residual distribution. Consequently, the precision exhibits exponential scaling with a base greater than 2, surpassing the efficiency of uniform quantization and demonstrating even stronger empirical performance.
For practical numerical implementation, the search for is conducted using Monte Carlo estimation of F(u) collected from the calibration set. Owing to the piecewise nature of F(V), we first initialize 0g using a coarse grid search, followed by refinement via the Newton iteration method:
0. = 02 2F(Ok) - F(0x+1) - 2p(0s) + p(0x+1)
(22)
where p(0x) is estimated by sampling within small intervals 8 around 0g, i.e.
F(0)' p(0))=p(0>>> ~
28 Σ
8 1
(23)
To further reduce quantization bias caused by Heaviside activation, we adopt a rounding strategy similar to [4] for the last neuron in each group. Specifically, we apply linear compensation to the input and modify the step function to be symmetric around the threshold:
0.5
((1) + q(() + OK+I,
(24)
thereby compensating for the flooring bias in spike generation.
6 Experiments
6.1 Implementations
Fission Encoding is post-training strategy broadly compatible with diverse SNN architectures. It leverages high-accuracy training with long time-steps while enabling low-latency inference at deploy- ment. We evaluate it across representative paradigms, including aantinuous-activatinn conversion Calib 28), quantized conversion QCFS, SRP [4] 17], direct training for spike CNNs TEBN [10], post- training conversion of ANN Transformers STA
24 and spike-based Transformers Spike-driven
49)
For classification, we conduct experiments on CIFAR-100 26 and ImageNet (6] using ResNet20 VGG16 (38), and spike- bakke-baedd Transformer
49 Converted models use T 32 time-steps, while

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Table 1: Performance comparison before/after Fission. Mem., Time and Energy per epoch are measured directly on-chip. All accuracy values reflect the best performance at evaluated time-step.
Time Fission Accu. Mem. Time/ Step
Energy (%) (GB) Epoch(s) (Wh)
Time
Fission Accu. Mem. Time/ (%) (GB) Epoch(s) (Wh)
Energy
Method
Method
Rate
Step
Rate
CIFAR- 100 & ResNet20
ImageNet & VGG16
32
76,32 No
56,8
0.64
32
63.64
604
7.55
Calib,
Calib,
16
64.48
29.1
0.37
16
No
55,79
3.71
302
4.30
0.85
(Conversion)
(Conversion)
8
32.10
16.3
0.22
28,10
154
2.37
20.43
10.0
0.14
16
0.15
61.94
3.93
324
4.75
16
0.10
76.20
0.88
31.4
0.42
Fission
0.87
60.79
5.28
169
2.61
8
0.78
73.28
1.17
19.5
0.27
1.57
58.31
6.21
88
1.43
Fission
4
1.35
73.11
1.31
12.0
0.18
QCFS.
32
68.10
627
7.82
2
3.70
70.50
2.25
6.4
0.11
No
3.54
(Quaniization)I6
50.97
324
4.49
32
76.78
59.6
0.70
16
0.23
67.32
3.87
332
4.61
QCFS.
16
No
67.54
0.81
30.2
0.36
Fission
8
0.91
64.51
5.01
157
2.91
(Quantization)
8
62.60
0.21
1.80
62.87
6.92
93
1.53
4
55.25
9.7
0.12
TEBN.
No
69.03
3.62
78
1.16
16
0.16
74.61
0.84
31.8
0.39
Fission
4
8
0.69
1.17
72.79
72.15
1.09
1.22
16.4
10.1
0.18
0.22
Fission
0.39
1.31
64.19
67,85
4.19
5.77
28
47
0.84
0.69
2
2.93
69.43
5.9
0.13
ImageNet & ResNet34
SRP.
32
65.50
55.1
0.57
32
64,82
2.68
348
3.02
16
64.71
28.3
0.35
Calib. (Conversion)
1
No
0.80
16
No
56,21
189
1.73
(uantiizaioon
8
62.94
15.7
0.18
31.75
110
0.95
59.34
8.0
0.11
16
0.17
66.94
2.81
214
2.01
16
0.28
68.72
0.86
27.8
0.39
Fission
0.96
63.75
4.60
135
1.38
Fission
8
1.02
66.41
1.13
14.4
0.22
1.72
60.43
5.57
55.2
0.61
2.71 4
65.39
1.65
9.1
0.16
QCFS.
32
69.37
2.48
372
3.29
TEBN.
4
No
76.13
0.83
9.8
0.18
(Quantization)16
No
59.35
228
2.08
(Training)
2
56.04
5.0
0.11
16
0.28
67.93
2.96
240
2.41
Fission
2
1.47
73.46
1.18
5.3
0.14
Fission
1.04
66.51
4.38
156
1.54
1
1.95
69.20
1.50
3.2
0.09
1.95
62.41
5.05
81.3
0.95
CIFAR-100
ViT-B/32 (ANN Transformer Conversion)
ImageNet Spike-driven Transformer 8-384
STA.
64
No
85,25
8.71
128.3
1.72
Spike-driven.
No
72.28
4.57
192
3,49
(Conversion)
32
84.15
70.5
1.03
0.54
72.41
5.54
98
1.91
Fission
16
0.53
82.70
10.4
39.2
0.77
Fission
0.98
70.29
6.49
60
1.02
8
1.15
80.28
15.7
24.3
0.42
directly trained models use T = 4. Fission Encoding is applied post-training with 2,000 calibration samples for CIFAR-100 and 20,000 for ImageNet. For image generation CIFAR-10. 10, we convert 4-layer spike CNN generator (WGAN-GP (3) and a U-Net diffusion model 19 both configured with T 64 and calibrated the full 60,000-sample dataset. The output layer remains in non- spiking RGB format. All models are deployed on a Lynxi HP201 neuromorphic accelerator (16GB), commercial successor Tianjic (34), for real-time measurement of memory usage, latency, and energy consumption. More details see Sec.
6.2 Main Results on Image Classification
Table |uummarizes classification results with Fission Encoding across various time-steps, training strategies, and architectures. We report top-1 accuracy, on-chip memory, inference latency per epoch, and energy usage, all measured on the Lynxi HP201 accelcrator with a batch size of 32. Fission Encoding consistently improves the accuracy-efficiency trade-off. By reallocating precision spatially instead of extending spike windows temporally, reduces inference time-steps by up to 8x while maintaining competitive accuracy. For example, decreasing T from 32 to 4 results in less than 5% accuracy drop, up to 83% energy savings, and only a 1.6x memory increase.
This trade-off is particularly advantageous for neuromorphic SNN deployments, where memory budgets are typically modest. The additional memory introduced by fission remains well below the 16 GB on-chip capacity of the HP201 accelerator, obviating the need for off-chip storage or compres-

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sion. Furthermore, the increased neuron count has negligible impact per-cycle power, while the substantial reduction in time-steps markedly lowers both latency and total energy consumption.
Figure illustrates how accuracy and power consumption scale with time-steps on ResNer20. At moderate T, only small fraction of fission neurons required to recover full accuracy. Under ultra- low latency constraints (I' 1-2), additional neurons significantly improve precision, maintaining competitive performance with minimal spatial overhead. These results demonstrate that Fission Encoding achieves exponential efficiency gains while keeping spatial complexity bounded. A more detailed analysis of memory and computational overhead is provided in Appendix
6.3 Main Results on Generation
Accuracy and Energy Consumption Different Time-stop
We further evaluate Fission Encoding image gen- cration tasks, which demand higher activation preci- sion due to richer decoding information flow. Con- ventional ANN-based generators typically require least 6-bit precision (corresponding to 64 time-steps
Higher Accuracy
70
Jocursiy Phi
Less Consumption
in SNNs), making low-latency inference challenging.
60
We convert two pretrained
[13]
T=16
and DDPM 19 intt SNNs using post-training cali- bration 28 and apply Fission Encoding with increas-
Tw2
Baseline
ing fission rates as T' decreases: [10, 30, 60, 90, 120]
Row Model
for 7-64 to T-4. Directly reducing T below 32 severely degrades image quality, whereas Fission En- coding recovers visually comparable results even at T=4, as confirmed by PSNR measurements against ANN outputs. Under low-latency settings, DDPM models outperform WGAN-GP preserving consis- tency despite reduced detail. To our knowledge, no prior work reports comparable visual quality < 8, enabling practical generative inference on neuromorphic hardware.
1/26
1/8
1/2
Energy Consumption
Figure 4: Accuracy and energy consumption across time-steps. Each point on the same curve denotes a different fission rate.
6.4 Further Analysis of Algorithm
Fission Rate and Threshold. Fig shows that accuracy initially improves rapidly with increasing fission rate and saturates once most layers achieve sufficient precision. At very low time-steps, the saturation point shifts due to the SNN unevenness effect [17, where excessive residual potential accumulates in late steps. Layer-wise analysis in Fig reveals that deeper layers typically undergo more fission ("Fission 1/2" denotes activations split into two or three neurons). Fig further illustrates that high-sensitivity neurons often exhibit long-tailed or multimodal activation distributions; optimal thresholds derived after fission (red lines) effectively capture these patterns.
A Ablation and Sensitivity- -Based Selection. We further examine the effect of Stage-1 sensitivity selection within the pipeline. Unlike prior coding schemes 24 30 that uniformly allocate precision, our approach selectively applies fission to highly sensitive neurons. As shown in Fig. g disabling selection (i.c., fissioning all neurons) slightly increases peak accuracy (< 2%) by including rare but important units. However, this significantly increases computational and memory overhead. With selection enabled, about 70% of neuron fission avoided, yielding a better acctracy trade-off.
Comparison with Baselines. We compare Fission Encoding with two ropulation-coding methods: Spatial Approximation [24] and Group Neuron 30) Even without sensitivity selection, our approach achieves 68. 22% accuracy with 100% additional neurons, surpassing the 66.94% of (24.. When each method uses its optimal neuron budget, baseline accuracy plateaus 72.85% with 400% overhead, while ours reaches 73. 28% with only 78%. These results demonstrate that Fission Encoding delivers both higher accuracy and significantly better neuron efficiency.
6.5 Computational and Hardware Implications
Computational Complexity. Transitioning from cascade to parallel firing introduces more compar- isons (CMPs) but significantly reduces additions (ADDs). For a group of n split neurons, achieving llevel precision requires 2 CMPs and ADDs with our approach, compared to 2" P1-1 CMPs

---

Te
Te32.
T-16
Teq
T-n
T=54,
T=32
Tw16
T
Twc
PSN8--26.97
P011-23-13
PSNR-19.26
PSNR-115.0 .0
PSh8
POUR
PSNR
PRAR
PSNB
Tw64, fw10%s
Tw32, 20%00%
T-16 fre60ok
T=B: Red0%
T-d. 20%.
To frw10%
Tw32, fre30%
T-16, fred0%.
Twd fr-d0%
Ted fnw120%
(a) WGAN
(b) DDPM
Figure 5: Image generation and PSNR for CIFAR-10 10 at varying inference time-steps. Top row:
original SNN outputs. Bottom row: results with fission encoding applied.
Fission Neurons each Layer ResNet20
Reature Disttioution and Rosion Thrichold Sampled RooNet20
Saturate Accuracy
Rosian
Teai teurens
Layen
166
1026
200-
332
134
Fixion Rate(Ka
400
Acc curarry 080
CA K
KG
3.12
E00
2.02
Nunoor
Layon 32
IF Layen 18
200-
17.90
200-
20
20c
50
No Selection
Sersetivity Selection
Figure 6: Neuron fission rounds Figure 7: Activation value distribu- across layers. Red numbers is the tions. Red line reflects the fission average fission rates per layer.
Figure 8: Comparison of accuracy and fission rate with related works and ablation of sensitivity selection.
thresholds and the response modes.
and 2 ADDs in conventional parallel coding. While CMP complexity grows exponentially in both cases, ADD operations operations-ttyicallly more expensive scale only linearly, making computation substantially more trrdware--riendly The slightly increased CMP cost is offset by exponentially fewer time-steps, yielding overall energy and latency benefits without hardware modifications. Further complexity analysis is provided in Appendix
Compatibility with Pruning. Fission Encoding can be combined with unstructured pruning to further reduce memory overhead by removing neurons with low sensitivity and weak post-synaptic weights. While less effective conventional GPUs, this sparsification provides clear gains on neuromorphic hardware: on ResNet20 with 22=2, pruning reduces memory usage from 2.25 GB to 1.53 GB with less than 2% accuracy loss.
7 Conclusion and Discussion
Activation quantization under limited time-steps remains a critical obstacle to scaling SNNs toward complex tasks. Drawing inspiration from post-training quantization in ANNs, reinterpret pop- ulation coding from a hardware-awan perspective and propose Adaptive Fission, a post-training mechanism that redistributes precision across the network while preserving binary spike compatibility.
Our design is motivated by a simple observation: conventional ANN quantization imposes two unnec- essary constraints- global fixed bitwidth and fixed bit weights (e.g.. powers of two). By representing precision through multiple spiking sub-ncurons, Adaptive Fission relaxes these constraints and flexi- bly allocates representational capacity where it is most needed. This enables efficient conversion of ANN backbones into SNNs without additional training or fine-tuning.
While broadly applicable across architectures and tasks, our framework shares limitations with existing ANN-to-SNN conversions. It is not directly suitable for event-driven datasets (e.g., CIFARIO 100 DVS), where input dynamics evolve with simulation time. Moreover, the temporal gains achieved by Adaptive Fission come at the cost of increased spatial complexity, which may raise resource demands in large-scale deployments. Despite these trade-offs, we believe that this precision-aware, post-training strategy represents a practical step toward bridging the gap between high-performance ANNs and efficient neuromorphic deployment.
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Acknowledgements
This work was supported in part by the National Key Research and Development Program of China under STI 2030 2030-Major Major Projects (No. 20212D02003000, in part by the National Key Research and Development Program of China (No. 2024YDL,0006, part by the National Natural Science Foundation of China (Grant No. 62176133), in part by the Tsinghua University Meituan Joint Institute for Digital Life under Agreement No. CO2 C0210322000380, in part by the Tsinghua- Data Technology Joint Research Institute (Project No. JIDT2024013) and in part by Qualcomm Technologies, Inc. under Statement of Work No.T51617560.
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19

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A Detailed Proofs and Explanations
This section provides proofs and explanations for theorems and propositions in Section 3 and
A.1 Explanations of Equations 5 and
We provide additional clarification on the motivation and interpretation of the two key equations used in ANN-to-SNN conversion preliminaries.
Weight Normalization for Hardware Compatibility, In the weighted-spike formula- tion introduced in Eq. 3 the spike output s(t) takes values in {0, instead of the binary set {0. 1) assumed standard IF neuron models. This discrepancy causes mismatch when deploying SNNs on neuromorphic hardware or when aligning their computation with standard ANN formulations.
Equation 5:
A common solution in training-free ANN-to-SNN conversion 228 2
is to absorb the threshold
scaling factor into the synaptic weights. For example, when a presynaptic neuron emits a spike with
magnitude ((--1) and the postsynaptic neuron fires if its membrane potential exceeds ( the firing condition th
vih- 1)
w(()
(
(25)
This inequality is equivalent to
v (6-11 W(()
th
1,
(26)
Vib (8)
indicating that the threshold (-(8-) can be absorbed into the synaptic weight, thereby fixing the threshold without changing the neuron dynamics.
Following the same principle, Eq. performs linear rescaling:
s(t) = ((((t)((()))))))))() - e {0,1}, {0.11 0
W == Ow, s(t)
(27)
which normalizes the spike outputs and ensures computational compatibility with hardware platforms that support only binary spikes. Conceptually, this step aligns the weighted-spike model with a standard (0, 1} IF neuron model, while preserving equivalence input-output behavior.
Equation o Firing-Rate Approximation as Quantized Activation. Equation 6 provides an interpretation of the firing rate of a spiking neuron in terms of quantized ANN activation. In rate-coding SNNs, the firing rate over 7 time-steps,
T
s())
(28)
represents the activation intensity. Assuming that the total integrated input matches the ANN pre- activation a, i.e., Ee Jsnn (1) = Jann = 0, the firing rate can be rewritten as:
a v (T)
0 clipp .0.7)
(29)
T
This expression shows that the firing rate effectively corresponds to a 7'-eeel quantized version of the ReLU activation The term [a/0] represents the discrete spike counts (bounded by 7), while the residual membrane potential u(T) reflects the quantization error. Such a formulation is widely used in quantization- conversion approaches (28 77 where the temporal accumulation of spikes treated as a discretized approximation of continuous activations.
Furthermore, Eq. 6 reveals useful design principle: since the quantization granularity is determined by a larger simulation window allows for a smaller threshold 0 Rs max(a)/T thereby enabling finer-grained representation of neuronal activation with minimal guantization error.
20

---

A.2 Additional Proof for Population Encoding Models
Proof Prop This is a special case of Prop 4.3 when the bit-length of neuron group k
See the proof of Prop 4.3
Proof for Prople 4.3
Define (())..... (m)) (()) E S(1) S .... S 8(m)
where m = (T + 1) as an ordered sequence
of all possible spike combinations. Consider ((x) E (5(=-1):8(++1) The errors within this interval can be calculated as:
(()))
( (((t)))
(())
(g ((-()-1))
(q (1-(())d
8(1-1)
)(1))
1
(() 2 (81(1-1) + 8(1+1))
+ [8(1-1) {(+1)}
($(1+1) -((i-)]
where the equation taken when S(1) = (8(1-1) + ((141))- Define 8(0) = 0, 8(m+1) = Imnxx as boundary conditions,
1
Σ 1=1
(()) + 22 [s(1)) - -s(o)) + ($m+1) -s(m)))
24max
1 84max
(2(8(1) - 8(0)) + (8(2) - 8(0)) + (8(m+1) 8m--1)) 2(8(m11) s(m))
4(8(m+1) 8(0)) )2
Sqmax
( 11+ + )
Ag (T + 1)a
8quax
Imax
2(T'
1)4
where the equation is taken when 8(4) (8(--1) + s((+1)) holds for all 1,.-., m. Thus, have
4max
$1 2(T + 1)
mmax
0, (T 1)¹
Proof for Prop 4.4 Consider multiple neurons operating at the final single step, setting T := 1 and Utemp (T)) Substitute these into E to obtain the above result.
A.3 Proof for Theorem 5.1
Proof. For clarity, we let fold 1 and denote 0x+1 8, consider v E [0, 8.) in most cases. The reduction caused by fission is given by:
0 < v <0
AU 8-
0
0 5 0
0 1 0 <ccl
21

---

Using the CDF to represent the piece-wise function, we can rewrite the error reduction sample batch. The optimization target is defined
EjAu] = Au p(v)do
-0 / 10s p(r))u -0) / /1-0s0s1 p(o)de
1 /
1-n p(u)do -1 (1- /-' p(v)de 1-0
F(1-0) F(0)) - F(1 0)] 0).
Given 0 = arg max, [[] the extremum can be obtained by derivation. At the maximum point,
we have:
dE Au
= [F - 0) F(0) - - 8 [2p(1 8) p(0) + p(1 8) = 0.
d0
where is the probability density function. Considering that in actual sampling, the distribution of approximately discrete, we assume p(0) = p(1 0) = 0. Thus we have:
2F - 0) (0) =0:0 EAv 1- F(1 - F(1 0) F(0)
yielding the result in Eq 18 We further estimate Ev by piece- scaling
0 <
vdv < OF(0),
1-8
8 [[F(1 8) 1-)((0) -
vdv S (1-8)) (L] 0) F(0)),
-1- () F(1 - -0) <
vdu I 1 F(1 - 0).
1-8
Substituting this to E(u] and [[Au]]
F(1 8) back to Eu can derive the range for [[Au].
Algorithm Overall Algorithm
Input: Original SNN; time-step T
Set fission rate f. required accuracy p%.
Collect batch of input data (i)
while Validation accuracy < p% do
for each feature dimension fea do
Conduct SNN forward propagation for T steps to calculate residual potential Ufea (T).
Conduct ANN forward and backward propagation to calculate gradient da
degut.
end for
Calculate sensitivity a with (T) and doous (cf. 16
dayen"
Determine threshold dsh. as the /-percentile all az for each layer / 1, 2.... m in the SNN do if sensitivity of i-th neuron du 2 Tth then
Perform fission encoding (cf. Theorem 5.1 and obtain new threshold sets O₃: of post-fission neuron group end if end for end while
Output: Threshold sets 0y for fission neurons
22

---

B Guidelines for the Overall Pipeline
We first provide the algorithm for our pipeline described in Section The fission rate f should be dynamically adjusted based on the network architecture and the desired reduction in time-steps. For modest reductions in time-steps (e.g., halving the time-steps), setting f Rs 15% is typically effective. For more substantial reductions, f can be incrementally increased by 15%, and further tuning may be required based on empirical performance. In the case of an ANN-to-SNN SNN conversion with 32 time-steps, where the inference requires at least time-steps (i.e., reducing to one-quarter of the original time-steps), a higher threshold of f zs 80% is usually sufficient. This can often be achieved with just one round of fission encoding. However, for more aggressive reductions, where fewer time-steps are required (e.g., reducing to 2 or steps), the fission rate f should be lowered to around 35%, and multiple rounds (typically 2 to 4 iterations) of fission encoding may be necessary to ensure performance convergence. The fission rate should be continuously adjusted based on the specific network architecture and the corresponding trade-off's between computational efficiency and accuracy. Empirical evaluation is recommended for fine-tuning these parameters in different settings.
C Spatial Complexity Analysis
We analyze the spatial and computational implications of neuron fission from four complementary perspectives:
Memory Overhead. Consider a fully connected layer Linear(n, n) with n thresholds, n membrane potentials, and n² synaptic weights. After fission with average rate k, each original neuron is split into k + 1 neurons. These fissioned neurons share the same input (membrane potential) but have independent outputs, which determines how overhead arises:
Thresholds: Increase from n to nkk + 1). one per fissioned neuron.
Membrane Potentials: Theoretically remain since inputs are shared, but in practice duplicated to n(k 1) for matrix computation efficiency.
Synapses: Each new neuron contributes n additional outgoing synapses, resulting in an increase of nk x n weights.
Thus, memory overhead grows linearly with the fission rate k across all components. In deployment, compiler optimizations such as in-place computation and buffer reuse typically reduce actual memory usage below this theoretical bound.
Computation Complexity (CMPs and ADDs). The transition from cascade to parallel firing indicators introduces additional comparisons but dramatically reduces additions. For a group of n split neurons, the parallel model requires 2 comparisons (CMPs) and one addition (ADD) for the i-th neuron, leading to a total of 2n CMPs and n ADDs for 2"-level precision. In contrast, achieving the same precision with conventional parallel scheme would require 2 neurons, each with one CMP and ADD, resulting in 2" 1 CMPs and 2" - 1 ADDs overall. While CMPs grow exponentially in both cases, ADDs - typically more expensive - grow only linearly our method. Although a cascade implementation would reduce comparisons to n, its sequential nature hinders parallel execution and is thus unsuitable for hardware deployment.
Hardware Implications. Threshold comparisons align well with neuromorphic architectures, where CMP circuits often implemented as dedicated units due inherent spiking dynamics. Comparators typically consume 3-5x less energy and have lower delay and area than adders. As a result, the CMP-dominant computation pattern introduced by Adaptive Fission is well matched to hardware execution. Moreover, per-step complexity increases are offset by exponential reductions in time-steps, yielding overall computational efficiency gains.
Power and On-Chip Efficiency. Measured power profiles show that a significant portion of neuro- morphic ASIC energy is consumed by I/O and memory access rather than arithmetic computation. Consequently, the growth in CMP operations has limited impact on total power consumption, while reduced time-steps translate into substantial energy savings. This trade-off slightly higher per- step complexity for significantly improved temporal efficiency - is generally advantageous for neuromorphic deployment.
Hardware Compatibility. All neurons, including fissioned ones, are implemented as modular IF units with segmented step functions expressed via CMP-based firing logic. Spikes remain binary
23

---

00 and are normalized to {0, during deployment. Hardware experiments on our prototype neuromorphic chip confirm that Adaptive Fission integrates seamlessly without requiring custom logic paths.
D Discussion on spike-based ViTs and LLMs
Our method is applicable to spike-based vision transformers, demonstrated with the Spike-driven Transformer in the main paper. However, many existing models still include floating-point operations (e.g., Softmax), limiting full compatibility with neuromorphic hardware. We believe that future spike-based transformers should avoid such nonlincarities to enable efficient deployment. We further tested our method on the Spatio-temporal Approximation model [24). boeating-point--ae ViT-B/32 backbone from CLIP, used for zero-shot classification. Results on both GPU and the HP201 ASIC are shown Table 2 While basic operations executed successfully, limited compiler support led to low execution efficiency, preventing meaningful power measurements.
Method
Time-Step Fission Rate
Accu. (%)
C1FAR-100 & CLIP-pretrained ViT-B/32 Zero-shot
64
85.25
STA
No
32
84.15
16
0.53
82.90
+ Fission
8
1.15
81.48
Table 2: Accuracy comparison of pretrained ViT-B/32 by CLIP for Zero-shot classification.
Applying method large language models (LLMs) faces additional challenges:
Attention scores typically require at least 8-bit precision, translating to long inference durations in SNNs. Adaptive Fission can help alleviate this, though memory costs may become more noticeable.
Attention precision:
Error accumulation:
The scale of LLMs amplifies small precision errors. More refined fission rate allocation strategies may be needed across network depth balance overhead and accuracy.
E
Experiment Settings
E.1
Hardware Resources
All our experiments were conducted on a workstation with Intel-13900KS, 32GB memory with a single 4090 GPU and a Lynxi HP201 neuromorphic accelerator. As commercial version of Tianjic chip 33,, HP201 adopts a many-core decentralized architecture, commonly utilized in neuromorphic chips, and integrates 60 configurable functional cores, 16 GB of memory, and image encoding/decodde units. This design allows it to support quantized ANN/SNN models of up to approximately billion parameters. The chip is packaged in a PCIe form factor with a peak power consumption of 55 W. The provided LynBIDI. library enables the compilation of SNN models based on Py Torch. However, since the underlying interfaces are not open, we are unable to directly measure the power consumption of individual cores or the number of MAC/AC operations during runtime. Instead, we record the total energy consumption using power readings from the sensors. While this method inevitably introduces interference (including power consumption from peripheral circuits, memory, and I/O interfaces), it reflects a more practical scenario. In such a case, reducing inference time significantly contributes lowering power consumption, the energy usage of peripheral circuits is largely determined by runtime duration.
Notably, while fission encoding can increase the number of neurons for single activation value, it does alter the network's original parameters. As result, the memory consumption typically increases by less than twice that of the baseline model.

---

E.2
Implementation of Classification
The baseline models and network architectures used in fission are directly sourced from the original codebase, including the Calibration [28] and QCFS 4 conversion methods, and spike-driven trans- formers 49. We slightly modify the TEBN [TO] hyperparameters suit ResNet20 and VGGI6, as the original experiments were conducted on ResNet19 and VGG11. Details can be found in the code.
E.3
Implementation of Image Generation
For the image generation model, we employ a 4-layer convolutional architecture based the improved Wasserstein GAN (WGAN-GP) [3]. The architecture structured as follows:
* Conv 1ran, 1024). BatchNorm2d, ReLU,
Conv TonnTssapposee21110 10024, 512). BatchNorm2d, ReLU,
Conv TranTrppppos2215122 256), BatchNorm2d, ReLU, Conv TranTrsppose22(556..3 3)
All convolutional transpose layers use a kernel size of 4. For the first layer, the stride is set to 1 and padding 0, while for the remaining layers, the stride is and padding 1. This model is trained on the CIFAR-10 dataset using the WGAN-GP framework [13]. After training, the model is converted spiking neural network (SNN) representation by discretizing the output into 64 time-steps using the Calibration (Calib.) method.
It is important to note that direct training of SNNs for image generation tasks, especially with lower quantization precision, typically fails to achieve satisfactory performance, as demonstrated by the difficulty in generating high-quality images. Consequently, methods such fission encoding employed to improve inference performance in this context.
For the diffusion model, we follow the implementation described by Ho et al. 19] The model consists residual blocks with a channel configuration of [2, 4, 4, 2]. and uses the Swish activation function instead of ReLU. This prevents direct conversion of the model to an integrate-and-Pee (IF) spiking neuron network. To address this, we introduce an IF layer after each Swish activation, thereby discretizing the continuous output into spike events.
The diffusion model is trained on CIFAR- with 500 iterations for image generation. After training. the model is also converted to 64 time-steps using Calib. However, it is worth noting that the ANN-based diffusion model requires 500 iterations for each image generation, which results in corresponding SNN model requiring 64 x 500 = 32,000 iterations. This computational cost is prohibitively high for real-time inference and necessitates the use of fission encoding to reduce latency.
For evaluating experimental results, we used the Peak Signal-to-Noise Ratio (PSNR) to measure the difference between the generated images and those from the original ANN model. The PSNR is calculated follows:
MAXX MSE
PSNR 10 logi0
(30)
where MAX represents the maximum possible pixel value the image data. For 8-bit images, MAX 255. The Mean Squared Error (MSE) is defined the average of the squared differences between the pixel values the generated image and the reference image, and is calculated as follows:
MSF
EEC.) K(
(31)
where ji,,j) and (iijj) represent the pixel values position (i.j) in the generated image and the reference image, respectively, and m and n denote the height and width of the images.
E.4 Random error and Reproducibility
Fission Encoding employs a small batch of examples for Sensitivity detection and estimates each neuron's cumulative density function, which introduces some variability. By allocating an additional validation set, we can assess the optimal execution, which is then applied to the test set. In most of our experiments, we selected the hest solution from runs on the validation set. Table dispplass the best and worst cases under these conditions, highlighting relatively significant performance variability.
25

---

stee
10%
20%
40%
80%
160%
320%
T=2
12.14-15.77
12.02-16.91
17.76-21.84
25.14-30.22
48.79-51667
60.43-66.55
T=4
14.06-22.65
18.10-24.3
27.69-31.74
50.67-57.69
67.35-72.80
70.44-74.73
T=8
35.51-38.70
36.32-42.53
52.69-57.40
68.11-73.26
69.32-72.54
70.68-73.07
T=16
68.65-71.43
72.74-76.10 73.31-76.21 73.55-75.90 72.42-76.79
74.51-75.84
Table 3: The best and worst accuracy range of fission rate at different time-steps within 10 runs.
F
Raw Data for Figures
In this section, we present the raw data corresponding to the figures in the main paper. The data points in Fig are provided by along with the supplementary Table [here. Table 3 corresponds to Fig
Time-step
10%
20%
40%
80%
160%
320%
T=2
15.77
16.91
21.84
30.22
51.67
66.55
T=4
22.65
24.38
31.74
57.69
72.80
74.73
T=8
38.70
42.53
57.40
73.26
72.54
73.07
T=16
71.43
76.10
76.21
75.90
76.79
75.84
Table 4: Accuracy comparison of fission rate at different time-steps. trained with 32 time-steps CIFAR-100 with calibration.
The raw model ResNet20
Method
Fission (%)
Accuracy
100
66.94
Group 30
400
71.67
100
69.39
Sensitivity + Group
150
72.85
100
52.51
Spatial 24
600
66.43
100
63.37
Sensitivity + Spatial
250
68.61
100
68.22
Fission (Ours)
200
72.40
100
72.79
Sensitivity + Fission
73.28
Table 5: Ablation and Comparison of different neuron selection and group combining methods.
G
Experimental Visualization and Analysis
This section provides detailed analysis of the various components of the fission encoding method, including demonstrations of its effects on neurons within actual networks, the distribution neurons across different layers after fission, and the distribution of sensitivity. Additionally, provide more results of image generation tasks high-resolution format.
26

---

Feature Distribution and Fission Threshold Sampled from Each Layer in ResNet20
11.3
Layorr
Lower
Layyon
Layen
3.70
Layor::3 30
Layer
12
Llayer
Layer: 24
10:01 20
0:50
Figure 9: Feature distribution and their fission threshold in ResNet20. The feature distributions presented are randomly sampled from the more sensitive neurons across various layers of the network. The horizontal axis represents the feature distribution, the left vertical axis (associated with the bar chart) indicates the distribution density, while the right vertical axis (corresponding the red line) reflects the fission threshold and the activation function resulting from the combination of neurons post-fission.
Fission Neurons in each Layer of ResNet20
Fission Neurons in each Layer of VGG16
Riszon Nourons
Nomocr of Ncurrrs
Number Neurors
40000
400.00
30000
$00.00
20000
50
20000
78 78
10000
10000
81 ND
B2
35
11
12
Figure 10: Distribution of Fission Neurons on all ReLU activation layers in ResNer20 and VGG16. when 30% of neurons undergo fission a single round. The distribution shows that although the number of activation values significantly decreases in the network, the number of fission neurons only slightly reduces, while their proportion increases in deeper layers.
27

---

Fission Neurons each Layer ResNet20
Fission Neurons each Layer VGG16
Total
60k
Rosion
60k
Fission Fisiion 2
Rosion
50k
50k
Number of Neurons
Number of Neursns
40x
40k
07
30k
30k
20k
20k
22
75
74
70
78
81
79
80
10c
10k
80
80
79
80
83
0K
0k
Layer
Layer
Figure 11: Distribution Fission Neurons across ReLU activation layers in ResNet20 and VGGI6 after two rounds of fission. The data indicates that the proportion of neurons undergoing multiple fission rounds is higher in the deeper layers, suggesting an increasing demand for feature precision in the deeper network layers.
28

---

Sensitivity Distribution of Each Layer in ResNet20
IF Layer:
Layer:
Layen 2
Layer:
Layer:
4007
186-
12000
1015
300
2010
24
IF Layer 5
Layer:
IF Layen
IF Layer: 8
Layan
2000
1500e
10000
156
3400
2000
02
Layer: 10
IF Layer: 11
Layer: 12
Layer:
11 Layer: 14
1508
1000
100
2014
2004
2000
2000
1900
Layen: 15
IF Layen 16
Layin 17
Layer:
4002
5007
3000
302
2064
0000
3e-5
1x-44
Sensitivity Distribution of Each Layer in VGG16
Layen:
IF Layer:
Layer::
IF Layer: 3
Layen
2550
2..0
1000
5000
5009
401C
1000
1004
200h
04
2.4
2.8
0.21
Layen:
Layer:
Layer:
Layer:
Layon
2040-
100C
40.0
10k
4450
18kC
659 0J8
IF Layer: 10
Layer: 11
IF Layer:
30-4
14-8
100
1800
608
15
65
15
Figure 12: Sensitivity in ResNer20 and VGG16. The distributions is characterized by long-tail distribution. The majority of neurons have sensitivity of zero, indicating that fission is unnecessary for them. However, due to variations in sample sampling, there may be some random error in this distribution.
29

---

T=64,
T=32,
T=16,
T=8,
T=4, PSNR=12.80
PSNR= 26.97
PSNR=23.13
PSNR=19.26
PSNR=15.80
T=64, fr=10%, PSNR=29.17
T=32, fr=30% PSNR=25.69
T=16, fr= PSNR=22.10
T=8. 9r= PSNR=20.04
T=4, 120%, PSNR=18.00
T=64, PSNR=26.24
T=32, PSNR=22.67
T=16, PSNR=19.49
T=8, PSNR=16.66
T=4, PSNR=14.11
T=64, fr=10%, PSNR=28.53
T=32, fr=30% PSNR=25.22
T=16, fr=60% PSNR=22.14
T=8, fr=90%, PSNR=20.80
T=4, fr=120% PSNR=19.17
Figure 13: Image generation results of WGAN and PSNR for CIFAR-10 different inference stme-steps.
30

---

T=64,
T=32,
T=16,
T=8, PSNR=11.90
T=4,
PSNR=28.66
PSNR=17.36
PSNR=13.82
PSNR= 5.86
T=64, fr=10%, PSNR=41.49
T=32, fr=30% PSNR=45.24
T=16, fr=60% PSNR=33.81
T=8. fr= PSNR=20.84
T=4, fr= PSNR=17.32
T=64, PSNR= 228.2
T=32, PSNR=17.19
T=16,
T=8,
T=4,
PSNR=13.85
PSNR-11.43
PSNR=5 5582
T=64, fr=10%, PSNR=42.04
T=32, fr=30% PSNR=41.47
T=16, fr=60%, PSNR=28.75
T=8, fr=90%, PSNR=20.14
T=4, fr=120% PSNR=19.07
Figure 14: Image generation results of diffusion U-Net and PSNR for CIFAR- 10 different inference sime-steps.
31


Reviews:
[{"review_id": "yFBPzBKZI7", "reviewer": "Reviewer_r9t1", "summary": "This manuscript introduces Adaptive Fission, a post-training population coding technique designed to reduce latency and power consumption in Spiking Neural Networks (SNNs) without sacrificing accuracy. Experimental results show significant improvements in performance and energy efficiency.", "questions": "What is the intuition or intuitive idea behind the adaptive fission proposed here?", "limitations": "The limitation is only briefly discussed regarding the structural complexity incurred by the proposed technique.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The main advantage is that this paper offers strong theoretical foundation and theorems supporting its design. The main disadvantage is that some of the theoretical derivations are hard to follow.\n\nHere are some detailed comments,\n1,There could be more explanation for the notations and derivative of some equations, such as equations (5) and (6). Besides, citations are needed here.\n2, There are dense notations in equation (15), (16) and (21), which make them hard to be understood; some intuitive explanation could help here. Also the number of the equation between (20) and (21) is missing.\n3, The discussion of sensitivity-based selection strategy can be expanded with its limitations.\n4. Some minor grammatical errors.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for detailed and constructive feedback. We address the raised concerns below and will incorporate the relevant clarifications into the revised version if possible.\n\n---\n\n### **Weakness 1: Clarification of conversion techniques in Equations (5) and (6)**\n\nThank you for pointing this out. Equations (5) and (6) follow established formulations in ANN-to-SNN conversion literature.\n\n- **Eq.(5)** is similar to scaling techniques in training-free SNN conversion, such as [28, Eq.5] and [45, Eq.5], which address the mismatch  between spike outputs in $\\lbrace 0, V_{th}\\rbrace$ model and normalized outputs in standard $\\lbrace 0,1\\rbrace$ IF model. For example, in [28]:\n\\begin{equation}\n\\mathbf{W}^{(\\ell)} \\leftarrow \\frac{V_{th}^{(\\ell-1)}}{V_{th}^{(\\ell)}} \\mathbf{W}^{(\\ell)}, \\quad V_{th}^{(\\ell)} \\leftarrow 1\n\\end{equation}\n*To illustrate: the previous layer firing is either $0$ or $V_{th}^{(\\ell-1)}$, and passes through a synapse with weight $\\mathbf{W}^{(\\ell)}$. The neuron receives $V_{th}^{(\\ell-1)} \\times \\mathbf{W}^{(\\ell)}$ as input and compares it to threshold $V_{th}^{(\\ell)}$ to decide firing. This comparison is equivalent to checking whether $\\frac{V_{th}^{(\\ell-1)}}{V_{th}^{(\\ell)}} \\mathbf{W}^{(\\ell)} > 1$, allowing us to absorb the threshold ratio into the weights and fix the threshold at 1.*\n\n  Similarly, in our **Eq.(5)**, since the spike output $s(t)$ in Eq.(3) lies in $\\lbrace  0, \\theta \\rbrace$, Eq.(5) linearly rescales $w$ and $s(t)$ via $w := \\theta w$ and $s(t) := s(t)/\\theta$, so that the computation is compatible with neuromorphic hardware supporting $\\lbrace 0,1\\rbrace$ spikes.  \n&nbsp;\n- **Eq.(6)** is a technique commonly used in quantization-based conversion methods (e.g., [28, Eq.9], [7, Eq.7]), reflecting the mapping between SNN spike counts and quantized ReLU outputs. This can be interpreted as approximating the discrete nature of SNN outputs ($\\lbrace 0, 1, ..., T\\rbrace$) via clipping and rounding operations in ReLU ANN. Its derivation can be found in the relevant citations.\n\n---\n\n### **Weakness 2: Intuitive explanation of Equations (15), (16), and (21)**\n\nWe appreciate your observation and agree these parts can benefit from improved intuition.\n\n- **Eq.(15)** describes how our neuron group transitions from a cascading to a parallel form. In Definition 4.5, we use a cascade-style explanation: higher-threshold neurons fire first, and any residual membrane potential is passed to lower-threshold neurons for further comparison. While conceptually clear, this mechanism is inefficient in practice.  \nInstead, we observe that the firing judgment for lower neurons can be **precompiled based on the total input, without waiting for upstream neuron activity**. For example, if two neurons have thresholds 2 and 1, the lower neuron can directly determine firing when the input falls in $[1,2) \\cup [3, \\infty)$, turing it into lightweight comparison logic. These firing rules are represented by our proposed indicator function.  \n&nbsp;\n\n- **Eq.(16)** estimates each neuron’s output sensitivity to residual error. It uses forward propagation to estimate **average residual potential** for each neuron, and approximated back propagation to derive **gradients of the output** relative to residual potential. Their product serves as a first-order approximation of how much residual error affects the final output.  \n&nbsp;\n\n- **Eq.(21)** gives a high-level summary of our optimal threshold derivation. The main text only presents a simplified form, and we refer the reviewer to Appendix A.2 for a complete proof. \n\n  In brief, the error function with respect to threshold $\\theta_k$ is step-wise and highly discontinuous. Even a small perturbation in $\\theta_k$ may lead to a large change in approximation error, making direct optimization of $\\theta_k$ extremely unstable.\n\n  To address this, we instead **represent the error with the cumulative distribution function $F$** of the original floating-point activation values. This function possesses more desirable properties — with sufficient sampling, $F$ is approximately continuous and differentiable. This enables us to estimate gradients with respect to $\\theta_k$ and derive near-optimal thresholds.\n\n---\n\n### **Weakness 3: Limitations of sensitivity-based strategy**\nThe inherit main challenge with the sensitivity-based selection arises when calibration samples are limited, leading to unreliable estimates. This issue has also been discussed in early ANN quantization works such as [r1-r3]. All these works emphasize the need for layer/channel-specific treatment due to varying sensitivity to quantization. \n\nNevertheless, their sensitivity analysis components are often relatively simple — typically relying on coarse statistics such as activation magnitude or gradient norms. This leaves room for further exploration of more fine-grained or theoretically grounded sensitivity estimation methods, particularly in the context of post-training adaptation.\n\n&nbsp;\n\n[r1] Quantization and training of neural networks for efficient integer-arithmetic-only inference. CVPR 2018.  \n[r2] Improving neural network quantization without retraining using outlier channel splitting. ICML 2019.  \n[r3] Smoothquant: Accurate and efficient post-training quantization for large language models. ICML 2023.\n\n---\n\n### **Question: The intuition**\n\nOur approach is motivated by the connection between SNNs and quantized ANNs, both restricted by insufficient precision. We observe that ANN quantization typically imposes two rigid structural constraints:  \n(1) a **fixed bitwidth** across all weights and activations (e.g., W4A4 assigns 4 bits uniformly),  \n(2) **fixed bitwise weight combinations** (e.g., binary weights with powers-of-two significance $1,2,4,\\dots$),  \nwhich are not inherently necessary in SNNs.\n\nThis motivates us to design an allocation scheme that allows both adaptive **bitwidth** and **bitwise scaling factors**, depending on neuron-level requirements. Since searching for a globally optimal bit allocation is combinatorially hard, we instead propose a sparse and iterative local strategy that incrementally allocates additional “bits” (i.e., spiking sub-neurons) only where needed. This insight forms the core of our adaptive fission mechanism.\n\n---\n\n### **Limitations: Discussion on structural complexity**\n\nWe discuss structural overhead in Appendix C and in response to Reviewer QkMA (Q2). As deployment overhead may depend on the hardware backend, we provide both algorithm-level spatial complexity analysis and real-world hardware measurements. We are open to provide further details on case studies."}, {"review_id": "kFEwnEsv5R", "reviewer": "Reviewer_hA37", "summary": "The paper proposes a novel method to improve information coding in spiking neural networks (SNNs), essentially for ANN-to-SNN conversions. Building on current population and encoding existing models, the technical contributions of the paper are as follows: 1) an algorithm to select neurons that require coding improvement, 2) a model that divides selected neurons into subpopulations of neurons and optimizes the neurons’ parameters to improve information coding.\n\nThe model was tested on neuromorphic hardware on image classification and image generation tasks and significantly improved energy consumption by reducing processing time, while maintaining competitive performance.", "questions": "I would also like to ask the authors to clarify if other existing methods use the cascade / parallel neural grouping models described in Section 4.\n\nAdditionally, I also noticed two minor typo: line 26 “needed” -> “need”, and line 43, “yield” -> yielding.", "limitations": "The authors properly addressed the different limitations of their work in the discussion section.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths: \n\n1) A sound and original model with clear mathematical foundations and which focuses on solving one well identified goal\n2) Strong evaluation of the work on multiple state-of-the-art datasets with a report on the accuracy, energy consumption and time processing performances compared to a control case/baseline network\n3) Valuable tests on neuromorphic hardware (Lynxi 48 HP201 neuromorphic chip) \n\nWeaknesses:\n\n1) The model requires adding more neurons to the networks to improve information coding. This adds an additional constraint on the memory usage and could be a bottleneck depending on the available resources/the hardware used\n2) The model is not compatible with event-driven platforms (e.g. event cameras) where input data vary at each timestep\n3) The model requires an additional training phase after conversion of the ANN to SNN or after training from scratch of the SNN", "was_revised": true, "final_justification": "The authors' rebuttal addressed correctly my concerns. I therefore do not decrease my score and maintain it to 5.", "rebuttal": "We sincerely thank the reviewer for highly positive and encouraging feedback on our work. We hope our responses can help further clarify the contributions and limitations of our paper. If accepted, we will correct the minor issues you pointed out in the final version.\n\n---\n\n### **Weaknesses**\nWe acknowledge the three limitations raised in the Strengths and Weaknesses section. To some extent, these are inherent trade-offs of our encoding strategy for inference acceleration. Since post-training techniques are often considered a \"no free lunch\" solution, our goal is to minimize the overhead while achieving substantial latency and energy improvements. Specifically,\n\n1. **Memory Overhead:**  \n   Adaptive Fission only selectively and sparsely introduces additional neurons to a small subset of high-sensitivity neurons, avoiding the redundancy caused by uniformly allocating bit-widths to all activations in conventional quantization.\n\n2. **Event-driven Compatibility:**  \n   We agree that the current method is more suitable for static inputs (e.g., images) rather than event-driven data streams such as event cameras. This is explicitly acknowledged in our discussion section, and is to be improved in future work.\n\n3. **Post-training Adjustment:**  \n   Regarding the additional training phase after conversion, we consider this to be more of a calibration step [28, r1, r2] rather than training, retraining or finetuning. This practice is also common in post-training quantization for ANNs [r3], and has two key properties:  \n   - **No weight update** is performed; instead, it adjusts a few parameters based on statistical properties extracted from a small calibration set, thus avoiding feature distortion or catastrophic forgetting.  \n   - It requires **very few samples** and typically no multi-epoch optimization, resulting in negligible overhead.    \n    \n    Hence, we believe this lightweight step does not impose significant practical cost.\n\n&nbsp;\n\n[r1] Error-aware conversion from ANN to SNN via post-training parameter calibration. IJCV 2024.  \n[r2] Adaptive calibration: A unified conversion framework of spiking neural networks. AAAI 2025  \n[r3] Accurate post training quantization with small calibration sets. ICML 2021.\n\n---\n\n### **Questions: Similar models in related works**\nTo the best of our knowledge, no prior work adopts the same form of \"cascade or parallel neuron grouping\" proposed in this paper. Some prior methods [24, 32] do involve neuron grouping but significantly differ in two key aspects:\n\n- **Limited representational power:**  \n  For instance, [24] uses multiple parallel neurons with uniformly divided thresholds to approximate multi-level firing. The $k$-th neuron is assigned a threshold scaled by $k/n$, and only one spike can be emitted at a time — yielding **linear** rather than **exponential** precision improvements.  \n  Similarly, [32] employs small auxiliary networks, but still lacks highly-independent binary combinatorial capability like our $\\mathbb{I}_i$ indicators (Eq. 15), again resulting in only linear improvements.\n\n- **Fixed weight allocation:**  \n  The combinations in prior works are either **uniformly preset** [24] or **learned via separate pre-training** [32], rather than dynamically determined based on neuron sensitivity as in our method."}, {"review_id": "jymvWWUJ88", "reviewer": "Reviewer_W46Z", "summary": "This paper proposes an ANN2SNN method, call ​adaptive fission​​, a post-training encoding technique for SNNs that ​dynamically allocates precision​​ by selectively splitting sensitive neurons into groups with different scales/weights, significantly reducing latency and energy consumption (up to 80%) without performance loss on neuromorphic hardware.", "questions": "**Question 1: Computational Cost of Parallel Model Indicators**  \nWhat is the computational overhead associated with generating the binary indicators $\\mathbb{I}_i$ in the Parallel Model implementation of neuron groups? Specifically, does the exponential growth in comparisons ($2^{i-1}$ per neuron $i$)?\n\n**Question 2: Impact of Sensitivity-Based Selection**  \nIf Stage 1 is not used to select sensitive neurons—and threshold fission is instead applied to all neurons—what would be the resulting accuracy implications and computational cost?\n\n**Question 3: Comparison with State-of-the-Art Methods**  \nThe proposed fission technique is not compared with state-of-the-art works, such as:  \nCNN Methods: \n- *Reducing ANN-SNN conversion error through residual membrane potential*  2023\nTransformer Methods:  \n- *SpikeZIP-TF: Conversion is all you need for transformer-based SNN*  2024\n- *Towards high-performance spiking transformers from ANN to SNN conversion* 2024  \n- *Spatio-temporal approximation: A training-free SNN conversion for transformers* 2024  \nI wonder what will be the result if the fission method is applied to the sota work?", "limitations": "See questions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:  \n1. **Theoretical Innovation**: Proposes Adaptive Fission—the first post-training method enabling dynamic bit-length/weight assignment with guaranteed exponential error reduction.\n2. **Hardware Efficiency**: Achieves 80% latency/power reduction via neuromorphic-compatible parallelization while maintaining binary spike compatibility.  \n3. **Flexible Deployment**: Compatible with ANN2SNN conversion and direct SNN training across classification/generation tasks.  \n\n**Weaknesses**:\nSee questions.", "was_revised": true, "final_justification": "I think the author address my concern compared to the SOTA experiments. I hope the authors should add the additional experiments to the new revision. Accordingly, I am raising my score.", "rebuttal": "We sincerely thank the reviewer for the thorough review and valuable comments. We respectfully hope these clarifications can help resolve your doubts and lead to a more favorable evaluation. Key results will also be incorporated into the final version and appendix if given the opportunity.\n\n---\n\n### **Question 1: Computational Cost of Parallel Model Indicators**\nYou are absolutely correct that our method introduces exponential growth in comparisons in exchange for finer control of spiking behavior. Despite this, this overall computational cost is still **exponentially more efficient** due to reduced additions, compared to conventional encoding schemes of parallel neurons.\n\n1. **Complexity analysis on algorithm**  \n   For a group of $ n $ split neurons, the parallel model involves $2^{i-1}$ comparisons (CMPs) and $1$ addition (ADD) for the $i$-th neuron, i.e., **$2^n-1$** CMPs and **$n$** ADDs in total, to achieve $2^n$-level precision.  \n   In contrast, to match this precision with conventional scheme, $ 2^n-1 $ neurons will be required, each with $1$ CMP and ADD, i.e., **$ 2^n-1 $** CMPs and **$ 2^n-1 $** ADDs in total, where **ADDs increases exponentially** compared to ours.  \n   Besides, the cascade version (Fig. 2 in the paper) requires only $ n $ CMPs and ADDs, but its sequential nature limits parallelizability and makes it unsuitable for deployment.\n\n2. **Hardware cost perspective**  \n   Neuromorphic chips often include dedicated CMP circuits due to threshold-based firing mechanisms. Comparators are typically 3–5× more energy-efficient than adders, with smaller delay and hardware footprint. Thus, the CMP-dominant nature of our method aligns well with parallel execution on neuromorphic hardware.\n\n---\n\n### **Question 2: Impact of Sensitivity-Based Selection**\n\nAblations on Stage 1 (sensitivity selection) can lead to the following differences:\n\n1. **Accuracy**  \n   Applying fission to all neurons (without selection) does not hurt performance and may slightly increase maximum accuracy (often $<2\\%$), since some rare but important neurons may otherwise be missed. But the added computational and spatial cost outweighs the marginal benefit.\n\n2. **Redundancy and energy**  \n   Without selection, each round of fission adds one neuron per activation value. After $ k $ rounds, the model expands by a factor of $ k+1 $. However, as shown in Figure 6, about 70% of neurons are never fissioned when selection is used. Fission on these inactive units results in negligible gain but increases memory and potential power cost.\n\n   Moreover, in Figure 8, focusing on the rightmost orange bars: full fission (no selection) with 200% additional neuron overhead achieves similar accuracy as sensitivity-based fission with only 60% overhead.\n\nThus, Stage 1 provides a cost-effective trade-off, concentrating computation on informative neurons without sacrificing accuracy.\n\n---\n\n### **Question 3: Comparison with State-of-the-Art Methods**\nWe appreciate the reviewer’s suggestion, and have thus extended our evaluation accordingly with the recent SOTA conversion methods mentioned [r1-r4]. We have previously reproduced both SRP [r1] and STA [r4], and now integrate our algorithm to them. \n\n| Method | Time Step | Fission Rate | Accu. (%) |\n|--------|-----------|--------------|-----------|\n| **CIFAR-100 & ResNet20** |\n|SRP($\\tau=4$) | 32        | No           | 65.50     |\n|  | 16        | No           | 64.71     | \n|  | 8        | No           | 62.94     | \n|  | 4        | No           | 59.34     | \n| + Our Fission | 16        | 0.28         | **68.72**     |\n|  | 8         | 1.02         | 66.41     |\n|  | 4         | 2.71         | 65.39     |\n| **CIFAR-100 & ViT-B/32(Supervised)** |\n|STA | 64        | No           | **85.25**     |\n|  | 32        | No           | 84.15     | \n| + Our Fission | 16        | 0.53         | 82.70     |\n|  | 8         | 1.15         | 80.28     |\n|  | 4         | 2.35         | 79.14     |\n\nThese results confirm that Adaptive Fission provides consistent improvements when applied to SOTA models. Since our method can be regarded as **post-training quantization**, it is broadly compatible with various pre-trained architectures.\n\n- For SRP[r1], which retains an ANN optimization target during conversion, the fissioned model can even **surpass the original SNN baseline** (e.g., at $T=32$) due to finer-grained encoding. \n- STA[r4], as a transformer-based method, is much harder with stricter precision requirements. As a result, more neurons are typically introduced during fission to preserve accuracy, leading to **slightly higher computational cost and accuracy degradation** compared to CNN cases.\n\nFor other 2 mentioned SOTA works:\n- **SpikeZIP-TF**[r2], as a quantized-conversion method, requires first retraining a quantized ANN prior to conversion. We are actively extending our method to support this framework.\n- **ECMT**[r3], as a follow-up of [r4], involves multi-threshold neurons. Extending Adaptive Fission to multi-threshold neurons is promising, but beyond the scope of current design assumptions of single-threshold. We are actively exploring this direction.\n\nBesides, due to the deployment complexity of non-CNN models on neuromorphic hardware, real-device benchmarks (latency/power) for transformer based architectures are currently underway and is to be included in future revisions.\n\n&nbsp;\n\n[r1] Reducing ANN-SNN conversion error through residual membrane potential. AAAI 2023.   \n[r2] SpikeZIP-TF: Conversion is all you need for transformer-based SNN. ICML 2024.   \n[r3] Towards high-performance spiking transformers from ANN to SNN conversion. ACMMM 2024.    \n[r4] Spatio-temporal approximation: A training-free SNN conversion for transformers. ICLR 2024."}, {"review_id": "uYGw83ZIgF", "reviewer": "Reviewer_QkMA", "summary": "This paper proposes training of a dynamic threshold for population encoding SNN model. By allocating a larger number of neurons to error-sensitive activation, they can reduce timesteps with minimized computation overhead. Demonstrating on a neuromorphic chip, this work reduces energy consumption on VGG16, ResNet20, and Transformer.", "questions": "1.\tWithin the implemented hardware, does it merge all $s_i(t)$ into $s(t)$ as in (14) and then pass the spikes to the next layer? Or, are all $s_i(t)$ in the group passed to the next layer in binary (spike) format, and instead the weights pre-multiplied by the thresholds $\\theta_i(t)$ are exploited on the synaptic operation of the next layer?\n\n2.\tCan you show how the increased memory overhead (threshold, potential, synaptic storage) is composed by the fission rate? I believe that such a quantitative analysis would further emphasize the paper’s strength.\n\n3.\tI recommend validating performance with ResNet-based models for the ImageNet benchmarks in Table 1. Presenting performance on ImageNet only with the very large VGG can diminish the contribution of this paper.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strength\n\n1.\tThis paper implements a dynamic number of neurons per activation, considering the saliency of activations. \n\n2.\tThis paper presents power consumption using the Lynxi HP201 chip. \n\n3.\tThis paper only needs calibration, not post-training. And their proposed training method is reasonable and can be adapted to various SNN works. \n\nWeakness\n\n1.\tAs this work uses multiple thresholds (neurons) per activation, the overhead from potential and output synapses becomes large. \n\n2.\tThe analysis with comparison baselines is insufficient. For example, group-neuron based SNN works [32] should be compared in terms of accuracy and energy. Furthermore, a comparison with A2SNN, which has reduced timesteps [1*] through post-training, may better illustrate the strength of this paper.\n\n[1*] N. Rathi and K. Roy, “DIET-SNN: A low-latency spiking neural network with direct input encoding and leakage and threshold optimization,” IEEE Trans. Neural Netw. Learn. Syst., vol. 34, no. 6, pp. 3174–3182, Jun. 2023.", "was_revised": true, "final_justification": "This paper proposes a novel method for grouping neurons that increases performance with minimized hardware overhead. This paper demonstrates the effectiveness of its approach by showing high accuracy in various benchmarks while demonstrating that the overhead incurred by actual neuromorphic hardware is not significant. Therefore, I believe that this paper presents strategies that various SNN models requiring accuracy improvement should consider.", "rebuttal": "We sincerely thank the reviewer for the detailed and constructive feedback. We hope our responses address your concerns regarding the validation of the proposed algorithm. We will include additional experimental results in the appendix if given the opportunity.\n\n---\n\n### **Weakness 1: Overhead of Additional Neurons**\n\nWe acknowledge that increasing the number of neurons per activation may introduce additional overhead. Since post-training techniques are often considered a \"no free lunch\" solution, our method is to **minimize** the overhead while achieving substantial latency and energy improvements. It only selectively and sparsely introduces additional neurons to a small subset of high-sensitivity neurons, avoiding the redundancy caused by uniformly allocating bit-widths to all activations and synapses in conventional quantization.\n\n---\n\n### **Weakness 2.1: Comparison with Group Neuron Baseline**\n\nWe provide comparative results with [32] in Appendix Table 5. For clarity, we summarize two key cases:\n\n- **Without sensitivity selection**, using the same 100% additional neurons, our method achieves **68.22%** accuracy, compared to **66.94%** for [32].\n- **With sensitivity selection**, allowing each method to use its own optimal number of neurons, [32] saturates at **72.85% accuracy** with extremely **400%** additional neurons, whereas our method achieves higher **73.28%** using only **78%** neurons.\n\nThese results demonstrate that our method is both more efficient and more accurate in leveraging population-based encoding.\n\n---\n\n### **Weakness 2.2: Comparison with Post-Training Methods**\n\nWe appreciate the suggestion and have initiated experiments to compare with DIET-SNN [1*]. Admittedly, there exists a fundamental gap of accuracy between our training-free approach and post-training methods like [1*]. The comparison experiments are still ongoing, and we provide some preliminary results.\n\nIn their original paper, the ANN accuracy on VGG16-ImageNet is reported as 70.08%, and the optimized SNN accuracy after post-training reaches 69.00% at $T = 5$.  \nSince their provided conversion strategy is not optimal for training-free, we adopt our own threshold selection method, and achieves an initial SNN accuracy of only **32.43%** at $T = 5$ — indicating a severe accuracy drop of training-free methods from the ANN baseline.  \nDespite this significant degradation, we were able to recover the accuracy to **62.84%** in our preliminary attempts, with **2.19×** additional fission neurons. Further experiments are still in progress.\n\n---\n\n### **Question 1: Spike Propagation and Equation (14)**\n\nYour understanding is accurate — the hardware implementation follows exactly the **second approach** you described, i.e., all $s_i(t)$ are binary and passed through individual synapses.\n\nEquation (14) is just a conceptual illustration, but since $s(t)$ is not binary, it cannot directly participate in spike-based synaptic operations. Actually, all split neurons in the fission group emit their binary spikes individually and independently to the next layer.  \nIn practice, we **duplicate** the synaptic weights associated with split neurons and **scale** them by their respective thresholds $\\theta_i$. These updated weights are compiled into an expanded matrix before deployment. This compilation step accounts for the main increase in memory overhead introduced by Adaptive Fission.\n\n---\n\n### **Question 2: Memory Overhead and Fission Rate**\n\nWe provide some brief conclusions in Appendix C, and a more detailed explanation here.\n\nConsider a $Linear(n, n)$ fully connected layer. The original layer contains:\n- $n$ thresholds,\n- $n$ membrane potentials,\n- $n^2$ synaptic weights.\n\nAssume the average fission rate $k$ is evenly distributed, i.e., each neuron is split into $k+1$ neurons. It is important to note that the fissioned neurons **share the same input** (membrane potential) but have **independent outputs**, which affects how overhead arises:\n\n- **Thresholds**: Increase from $ n $ to $ n(k+1) $, one for each fission neuron.\n- **Membrane Potentials**: Though theoretically $ n $ (since all split neurons share the same potential), we allocate $ n(k+1) $ potentials in practice with duplicated channels for shared values to achieve efficient matrix operations.\n- **Synapses**: The $nk$ newly created neurons retain the same inputs (since inputs are not split) but produce distinct outputs. Therefore, each of the $nk$ neurons still contributes $n$ additional synapses to the next layer, resulting in an increase of $nk \\times n$ synaptic weights.\n\nThus, the overhead scales **linearly** with the fission rate $ k $ across all components.\n\nIn real-world deployment, the actual memory growth is typically **less than the theoretical bound** due to hardware/compiler optimizations such as in-place computation and reuse of intermediate buffers. Further analysis on memory occupation component may involve an in-depth study on the compilation process.\n\n---\n\n### **Question 3: Evaluation for ResNet on ImageNet**\n\nWe have added **ResNet-34 experiments** on ImageNet using both Calibration and QCFS. Preliminary results show comparable or better improvements than VGG16, with similar fission rates.\n\n| Method             | Time Step | Fission Rate | Accu. (%) | Mem. (GB) | Time/Epoch(s) | Energy (Wh) |\n|--------------------|-----------|--------------|-----------|-----------|----------------|-------------|\n| **ImageNet & ResNet34**                                                                              |\n| Calib. (Conversion)| 32        | No           | 64.82   | 2.68      | 348            | 3.02        |\n|                    | 16        |              | 56.21     |           | 189            | 1.73        |\n|                    | 8         |              | 31.75     |           | 110            | 0.95        |\n| **+ Fission**      | 16        | 0.17         | **66.94**     |2.81      | 214            | 2.01        |\n|                    | 8         | 0.96         | 63.75     | 4.60      | 135            | 1.38        |\n|                    | 4         | 1.72         | 60.43     | 5.57     | **55.2**         | **0.61**    |\n| QCFS. (Quantization)| 32       | No           | **69.37**   | 2.48     | 372           | 3.29       |\n|                    | 16        |              | 59.35     |           | 228            | 2.08        |\n| **+ Fission**       | 16        | 0.28         | 67.93     | 2.96     | 240            | 2.41        |\n|                    | 8         | 1.04         | 66.51     | 4.38      | 156            | 1.54      |\n|                    | 4         | 1.95         | 62.41     | 5.05      | **81.3**         | **0.95**    |\n\nDue to time constraints, directly trained SNNs are still under evaluation. Despite this, sensitivity maps and per-layer fission distributions (Appendix Figures 10 & 11) show consistent trends across VGG and ResNet, suggesting strong generalizability.\n\nAs an interesting note, although VGG16 appears larger, the convolutional backbone has **fewer parameters** than ResNet-34 (14.7M vs. 21.3M). Most of VGG’s size stems from its final fully connected layers (~123M parameters)."}]

Author rebuttal:

Meta-review: This paper proposes a post-training encoding method that converts ANNs to SNNs, substantially reducing the number of SNN inference steps. Initial reviews were mixed, but after the rebuttal all reviewers recommended acceptance. The reviewers highlighted the method's solid theoretical foundation and strong empirical results, particularly the evaluation on real hardware. The AC concurs with the reviewers' post-rebuttal assessment and recommend acceptance. The AC encourages the authors to incorporate the rebuttal materials into the camera-ready version.
Decision: Accept (poster)