Differential Coding for Training-Free ANN-to-SNN Conversion
Authors: Zihan Huang
Wei Fang
Tong Bu
Peng Xue
Zecheng Hao
Wenxuan Liu
Yuanhong Tang
Zhaofei Yu
Tiejun Huang
Venue: ICML 2025 poster (2025)
Keywords: Spiking Neural Networks
ANN-to-SNN Covnersion
Differential Coding
Threshold Iteration
Primary area: applications->neuroscience_cognitive_science

Abstract: Spiking Neural Networks (SNNs) exhibit significant potential due to their low energy consumption. Converting Artificial Neural Networks (ANNs) to SNNs is an efficient way to achieve high-performance SNNs. However, many conversion methods are based on rate coding, which requires numerous spikes and longer time-steps compared to directly trained SNNs, leading to increased energy consumption and latency. This article introduces differential coding for ANN-to-SNN conversion, a novel coding scheme that reduces spike counts and energy consumption by transmitting changes in rate information rather than rates directly, and explores its application across various layers. Additionally, the threshold iteration method is proposed to optimize thresholds based on activation distribution when converting Rectified Linear Units (ReLUs) to spiking neurons. Experimental results on various Convolutional Neural Networks (CNNs) and Transformers demonstrate that the proposed differential coding significantly improves accuracy while reducing energy consumption, particularly when combined with the threshold iteration method, achieving state-of-the-art performance. The source codes of the proposed method are available at https://github.com/h-z-h-cell/ANN-to-SNN-DCGS.

Differential Coding for Training-Free ANN-to-SNN Conversion
Zihan Huang'
Wei Fang
Tong Bu'
Peng Xue 34 Zecheng Hao
Wenxuan Liu
Yuanhong Tang
Zhaofei Yu 15 Tiejun Huang'
Abstract
1. Introduction
Spiking Neural Networks (SNNs) are sometimes regarded the third generation of neural network models (Maass, 1997) for their unique neural dynamics and high biological plausibility (Gerstner et al., 2014), making them a compet- itive candidate to Artificial Neural Networks (ANNs) (Li et al., 2024). A significant difference between ANNs and SNNs is the information representation. ANNs transmit dense floating values between layers. While in SNNs, com- munications between layers are based on sparse and binary spikes, which are triggered by the membrane potentials of spiking neurons across the threshold, bringing event-driven computations and extremely low power consumption on neu- romorphic chips (Merolla et al., 2014; Davies et al., 2018; DeBole et al., 2019; Pei et al., 2019).
Spiking Neural Networks (SNNs) exhibit sig- nificant potential due to their low energy con- sumption. Converting Artificial Neural Networks (ANNs) SNNs is efficient way achieve high-performance SNNs. However, many con- version methods are based rate coding, which requires numerous spikes and longer time-steps compared to directly trained SNNs, leading to increased energy consumption and latency. This article introduces differential coding for ANN- to-SNN conversion, novel coding scheme that reduces spike counts and energy consumption by transmitting changes in rate information rather than rates directly, and explores its application across various layers. Additionally, the thresh- old iteration method is proposed to optimize thresholds based on activation distribution when converting Rectified Linear Units (ReLUs) to spiking neurons. Experimental results on ious Convolutional Neural Networks (CNNs) and Transformers demonstrate that the proposed differential coding significantly improves accu- racy while reducing energy consumption, par- ticularly when combined with the threshold eration method, achieving state-of-the-an per- formance. The source codes of the proposed method available at https://githu: com/h- z h- to
However, the discrete and non-differentiable spike firing process causes huge learning challenges in SNNs. Recently, this issue is solved partly by the surrogate learning method (Neftci et al., 2019), which redefines the gradient of spike firing process by smooth and differentiable surrogate func- tion. Enabled by the surrogate gradients, deep SNNs can be trained by powerful backward propagation and gradient de- scent methods, and their performance are greatly improved (Fang al., 2021; Duan et al., 2022; Shi al., 2024). The applications of SNNs are also extended to complex event- based vision tasks (Cordone et al., 2022; Liu et al., 2024; Chen et al., 2024; Liu et al., 2025). Unfortunately, the sur- rogate gradient method is a coarse approximation, and may mislead the gradient descent direction in multi-layer SNNs (Gygax & Zenke, 2024). The time dimension of SNNs leads the employment of backpropagation through time (BPTT), which requires nearly T times of training resources than ANNs. Here T is the sequence-length, and is also the number of time-steps of SNNs. Although some online learning methods (Xiao et al., 2022; Bohnstingl et al., 2023; Meng et al., 2023; Zhu et al., 2024) can estimate the full gradients of BPTT by accumulation of single-step gradients, their task accuracy is sub-optimal.
'Scoool of Computer Science, Peking University, Beijing, China "School of Electronic and Computer Engineering, Shen- zhen Graduate School, Peking University, Shenzhen, China 'Peng Cheng Laboratory, Shenzhen, China "Shenzhen Institute of Ad- vanced Technology, Chinese Academy of Sciences. Shenzhen, China "Institute for Artificial Intelligence, Peking University, Bei- jing. China. Correspondence to: Wei Fang <fwei pku.edu.cn.. Yuanhong Tang <ydtang @ pku.edu.cn>.
In addition to the surrogate gradient methods, the ANN to SNN conversion methods (Cao et al., 2015; Han et al., 2020; Li et al., 2021; Deng & Gu, 2021; Bu et al., 2022a; 2024) are another spiking deep learning methodology that
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Differential Coding for Training- Freen ANN-to-SNN Conversion
eliminates training challenges of SNNs. They convert pre- trained ANNs SNNs with replacing nonlinear activation functions by spiking neurons. The converted SNNs enjoy high performance and close accuracy to the source ANNs, even in the complex ImageNet dataset. Most of the conver- sion methods are based on the rate coding, which represents activations in ANNs by the firing rates of SNNs. However, the precise estimation of firing rates requires a large number of time-steps, resulting in the obviously higher latency and energy consumption of the conversion methods than the surrogate gradient methods.
method (Cao et al., 2015) was derived. As the firing rate is defined by the average number of spikes over all time- steps, its range is restricted between zero and one. For the negative part, the IF neurons perfectly fit ReLUs. While the outputs of ReL.Us are unbound, the normalization of weights for regulating activations (Rueckauer et al., 2017) and the balancing of thresholds for spiking neurons (Han et al., 2020) are proposed and relieve the range of mismatch during conversions.
The time is discretized to time-steps in SNNs. Consequently, the firing rates are also rounded with the fixed interval. While the floating activations in source ANNs are contin- uous, the discrete firing rates can not fit them preciously, causing the quantization errors. To future reduce the con- version errors, some quantized ANN to SNN methods are proposed (Bu et al., 2022b; Hu et al., 2023). These methods quantize and clip activations of ANNs, reliving the quantiza- tion errors and range mismatch at the same time. However, the source ANNs must be re-trained, which increases con- version costs, and their performance declines due to the change in the activation function.
In this article, we propose the differential coding and its im- plementation scheme for different layers in ANN-to-SNN conversion, Instead of considering the average firing rate as the encoded activation value, differential coding treats time- weighted spikes as corrections to the encoded activation value. This approach not only improves network accuracy but also allows neurons to stop firing once certain ap- proximation precision is achieved, thereby reducing energy consumption without any extra training. Additionally, by minimizing the expected error between the Rectified Linear Units (ReLUs) and the encoded values in SNNs, we pro- pose the threshold iteration method to determine the optimal thresholds for the spiking neurons for converting ReLUs, further enhancing the performance of the SNN.
The spikes may not arrive evenly during inference, which may cause the unevenness error in conversion (Bu et al., 2022b). A typical case is that no spike during the first _ time-steps, and more than I spikes from different synapses arrive at the neuron during the last T time-steps. Although the imput firing rate larger than 0.5, the neuron can not generate the 0.5 output firing rate because it has not enough time-steps to fire. Several methods have been proposed to reduce this error, including the two-stage inference strat- egy (Hao et al., 2023a) and shifting the initial membrane potential (Hao et al., 2023b).
Our main contributions are summarized as follows:
2
We propose differential coding for ANN-tO-SNN con- version and establish the dynamics for various modules in SNNs.
We design a threshold iteration method to determine the optimal thresholds of spiking neurons for converting ReLUs.
Recent research has been extended to convert ANNs with activations beyond ReLUs to SNNs. (Oh & Lee, 2024) introduced sign gradient descent based neuron that can approximate various nonlinear activation functions. (Wang et al., 2023) and (kang you et al., 2024) trained modified Transformers and converted them into spiking Transformers. Meanwhile, (Jiang et al., 2024) and (Huang et al., 2024) de- veloped modules to approximate nonlinear layers, enabling training-free free conversion of Transformers SNNs.
We designed two equivalent implementations of the employed tulti-threshold (MT) neuron to facilitate frriwayy execution.
By converting different CNNs and Transformers into SNNs for evaluation, our extensive experiments demon- strate that the proposed method achieves state-of-the- art accuracy while significantly reducing network en- ergy consumption.
2.2. Temporal Coding Conversions
2. Related Works
The rate coding method is inefficient and causes huge la- tency in rate-based ANN to SNN conversion methods. The surrogate gradient methods avoid this issue by the end-to- end training, while the interpretation of coding methods in these SNNs is not clear yet (Li et al., 2023). For the conversion method, manual design for the coding strategy is indispensable. Beyond the rate coding method, several temporal coding methods are explored.
2.1. Rated-based ANN to SNN Conversion
The rate coding method has been early found in biologi- cal neural systems (Adrian, 1926) that stronger stimulation causes more frequent spikes. This straightforward coding method builds the bridge between activations of ReLUs in ANNs and firing rates of Integrate-and and-Free (IF) neurons in SNNs, based on which the primary ANN SNN conversion
2

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Differential Coding for Training-Free ANN-1O-SNN Conversion
The time-to-first-spike (TTFS) coding method (Rueckauer & Liu, 2018; Zhang al., 2019; Stanojevic al., 2023) encodes the value into the firing time of spikes. Each neuron only fires one spike in TTFS SNNs, which brings extremely high power efficiency. However, these methods rely on layer-by-layer processing, i.c., one layer can only start to compute after it receives all input spikes from the last layer. Thus, these TTFS SNNs suffer from high latency increasing with the network depth.
Soma
Axon
Synapee
Dendriee
r(T-1)
w's!
P(T)
ma(r)
M
W
(7)
b/
d(T)
Figure I. Diagram of the MT neuron. The MT neuron receives input from last module and emits up to one spike at each time-step.
The phase coding methods (Kim et al., 2018; Wang et al., 2022b) are similar to binary/decimal conversion. They en- code values from spikes with the power-of-2 weights. For a given number of time-steps T, these methods can repre- sent 2 different values, while the rate coding method can only represent I' values. Their drawbacks are similar to the latency problem in TTFS SNNs that the values can only be obtained after all weighted spikes in phase arrive.
t. The dynamics of the MT neurons are described by the following equations:
m [t] v' 1] + [(] = r'it - 1] [t)
(2)
[t] = MTH (m [t].
(3)
x [t] Σ [s X/.
(4)
The burst coding methods (Park et al., 2019; Li & Zeng, 2022; Wang et al., 2025) imitates the bursts of spikes during a short period in biological neural systems. The burst spikes are implemented by the multiplication of spikes and a co- efficient, which carry more information than binary spikes, but may lose the advantages of SNNs based on the binary characteristic.
w'(t] = m [t] x (ttt)
(5)
0.if Azn < x < An
MTH.... (())))))
1.elif i == arg min, x Apl
0,else
(6)
3. Preliminaries
Figure shows the dynamics of MT neurons, when n 1, this model reduces to an IF neuron with an additional negative threshold. Since only up one threshold can emit spike per time-step, and X' can be derived by bit-shifting 6, we can implement MT neurons by calculating W'X through the weight W' followed by bit-shifting.
3.1. Multi-Thresholl Neuron
Many previous works have proposed using ternary-valued neurons to simulate negative values and reduce conversion errors (Li et al., 2022; Wang et al., 2022a; kang you al., 2024). The ternary representation, with outputs -1, O, and 1, does not disrupt the event-driven nature and significantly enhances the expressive capability. Furthermore, (Huang et al., 2024) introduced the use of multi-channel methods to implement multi-threshold (MT) neuron for spike commu- nication. In this article, we similarly adopt this approach to simulate y = x using identity spiking MT neurons.
3.2. Rate Coding in ANN-to-SNN Conversion
Traditional ANN-to SNN conversion methods employ rate coding, which can be mathematically expressed as:
[4]
x [c],
=
The MT neuron is characterized by several parameters, in- cluding the base threshold 8, and total of 2n thresholds, with positive and n negative thresholds. The threshold val- ues of the MT neuron are indexed by i, where X! represents the i-th threshold value in the layer l:
where ai] represents the encoded output of layer / in SNNs at time-step i, while r'[t] denotes the encoded activation that aims to map activation value a' from the corresponding ANN layer. Derived from Equation (2) and (5), we have
o' [t] r'it 1] = x'- [t]
x [t].
(8)
g¹
8
X/ = 0², 1²1²/1-
2 ,..
An
21
0¹
0'
o' [1] [0)
= x
[1]
x
14
Xh+1 -0,, No,2 2
Nen
1=1
1=1
2
1
[[t]
Let variables ['] W' st[.]. sent the input current, weight, the output spike of the i-th threshold, the total output signal, and the membrane poten- tial before and after spikes in the 1-th layer at the time-step
x' [t]. m m'[t], and o'[t] repre-
r't =
t x [t]
x
[t)
t
im
(10)
v'[(]
-
[0]
= [1]
3

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Differential Coding for Training-Free ANN-to-SNN Conversion
Assuming [1) _'11, and given that a' o'-1 when simulating y x. when is sufficient large v close to o' [0]. the encode value r' [c] in SNNs can approximate the activation value a' in ANNs:
relationship between the two is expressed by Equations (13) and (14), follows:
e' [t] = r'[t - 1] + x x'(t),
(13)
r 14 = r'1t - +1]+ x" [t]
111 = rf 11 v'1(] o [0]
(14)
t
t
(11)
rr''1(1 [1] = a²-1 = a².
where / starts from 1, [0] -0.
4. Method
The detailed explanation of Definition 4.1 is provided in the Appendix B. Comparing Equation (7) with (14), the key difference is that differential coding only updates the encoded activation value when an output spike occurs, rather than decay at each time-step in rate coding. Figure 2 shows the ideal fitting results of rate coding and differential coding for Input y = x with T 3 and thresholds 11. Differential coding can represent a wider range of values and achieve higher precision than rate coding, given the same threshold and time-steps
In this section, we propose differential coding with graded units and spiking neurons (DCGS), a training-free theory for converting ANNs to SNNs. We begin by introducing a differential coding approach, from which we develop differ- ential graded units, differential spiking neurons and differ- ential coding for linear Layer. These enable the conversion of various network modules. Additionally, we provide the threshold iteration method to find the optimal threshold of spiking neurons for converting ReLUs. The overall algo- rithm can be found in Appendix A. Furthermore, we design two equivalent implementations of the MT neuron to sup- port hardware- friendly execution.
Closest Encoded Value (3
2
Rate Coding Differential Coding
4.1. Differential Coding in ANN-to-SNN Conversion
The traditional ANN-to-SNN conversion uses rate coding to transmit information, where the firing rate [1] at each time-step encodes the activation value. Equation (12) shows the relationship between the output firing rate r' and the output signal x! [t). When layer consists of spiking neurons, the state can be described as x² [4] = 0's'( [4]. where s' [t] denotes the spike at time-step and represents the threshold.
-2
-2
0
2
Ideal Encoded Value
Figure 2. Comparison of idcal fitting results: rate coding vs. dif- ferential coding for input = x with T = 3 and thresholds +1. Differential coding shows wider representation range and higher precision.
r'(t]
Σ
1-1 r'
x! 1]+
(12)
t
When the neuron does not emit spike, the rate update is the proportion - -rr'(t when the neuron emits a spike, the rate update increases by -1r' - 1] =1([]
of the previous rate, while
4.1.1 DIFFERENTIAL GRADED UNITS
Existing ANN-to-SNN conversion methods struggle with nonlinear functions such as Gaussian Error Linear Units (GeLU) (Hendrycks & Gimpel, 2023) and LayerNorm (Lei Ba et al., 2016). For these nonlinear layers, we utilize specific neuron dynamic units to implement them. Based on the expectation compensation idea from (Huang al., 2024), we propose introducing differential graded units to replace those nonlinear modules that cannot be directly converted.
However, the rate coding method has problem: over time, the encoded value gradually decays. As the time-step increases, the influence of earlier inputs becomes smaller, and the system requires more spikes to compensate for this decay effect, thus increasing the number of spikes required.
To address this issue, we propose a novel encoding scheme, referred to as differentiaal coding.
Derived from differential coding scheme in Definition 4.1, this article proposes two types of differential graded units. Theorem 4.2 corresponds to nonlinear layers with only one input q/''((), and Theorem 4.3 applies to certain operations with two inputs x// [t) and 1.11 [t).
Definition 4.1. In differential coding, denote æ'(t] as the actual output of the neuron. Define e'[t] as the encoded output value at time-step t, the encoded activation value r' as the average of e'[{] from time to time-step f. The

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
Theorem 4.2. Let F be a nonlinear layer with only one
imput a! (f), such as Gelu, Silu, Maxpool, LayerNorm, or Softmax. In ANN-to-SNN conversion, the mapping from F to dynamics of the differential graded unit in differential coding is given by Equations (15) and (16).
Lincar
W
Lincar
W
b
Graded Unit
Full Spiking Neuron
Nonlinear
m' [t] = 11- = m'lt +]]+ x [(]]
x* (F Flim 113)
(15)
1
xe [t] t (m' F' (mm' 1)))
(16)
variables
Identity Spiking Neuron
where m'(1) [1] is the membrane potential at time-step which equal b'-1 if the previous layer has bias else O, r' [t] is the encoded output activation value of the previous time-
float
spikes
(b)
steps. The output of layer 1 at time-step which serves the inpul to layer 1 + 1, is given by xr (1)...
Full Spiking Matrix Product
Identity Spiking Neuron
Identity Spiking Neuron
172
m2,
The proof of Theorem 4.2 is detailed in the Appendix C. From Theorem 4.2, single-input unit requires two vari- ables: one to record m and another to record F(m'(()) in order to reduce redundant calculations at each time-step.
[A
Graded Matrix Product
J-1
will
m211 =lp
Theorem 4.3. Let be an operation with two imputs, such as matrix multiplication or element-wist wise multilicattion. In ANN-1o-SNN conversion, the mapping from operation to dynamics of the differential graded units in differential cod- ing is given by Equations (17) to (19).
Figure 3. (a) Conversion of a linear layer followed by a nonlinear layer in an ANN into SNN modules. (b) Conversion of matrix product or clement-wise multiplication in the ANN into SNN modules.
x [1]
m'Al'
-
[4] m'x 1] +
(17)
t
x [
mb((t)
[t]
1-1 [t] m'Ble A
1]
(18)
neuron in rate coding into a differential neuron in differential coding.
t
[t] 2/-01
x'((]
(+)
m'
+ mlx()
(t).
Theorem 4.4. In rate coding. the output of the previous layer, -'' is directly used the input current for the current layer [' = -') In differential coding, the input current [' can be adjusted as shown in Equation
t
(19)
where m'x(t] and m'n(t) are membrane potential at time-
step i, and r and " are the encoded activation values of the previous layers at time-stept t. The output of layer I at time-step t, which serves as the input to layer 1, is given by x² [t].
(20), which converts any spiking neuron into a differential spiking neuron:
I' ['(t] = m it] + x²
x' [t].
(20)
x
The proof of Theorem 4.3 is detailed in the Appendix D. From Theorem 4.3, neuron with two inputs requires two variables to record m', [t] and m'I [t). respectively. Graded units provide the ability to integrate information about non- linear layer changes. This enables the conversion of various complex networks, including CNNs and Transformers.
m + 1] 1]=m||[1]
[t] +
(21)
t
where m [0] is b' the previous layer has bias else 0.
The proof of Theorem 4.4 is detailed in the Appendix E. In contrast to rate coding, which is constrained by a decay that limits the output range to below the threshold 6, differen- tial coding allows for adaptive adjustment of the neuron's output range by directly modifying the encoded activation r' [4]. This flexibility is especially beneficial in scenarios with multiple or dynamically adjustable thresholds, as the combination of different thresholds enhances the represen- tation accuracy. So, we employ a differential version of identity tulti-th spiking neuron in our experiments.
4.1.2. DIFFERENTIAL SPIKING NEURONS
Since the majority of computations occur in fully connected layers, convolutional layers, and matrix multiplication lay- ers, it is recommended to introduce spiking neuron layers before these layers, so that the computation is event-driven, thereby effectively reducing the network's 's energy consump- tion. Theorem 4.4 demonstrates how to convert a spiking
5

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Differential Coding for Training-Fres Free ANN-1O-SNN Conversion
4.1.3. DIFFERENTIAL CODING FOR LINEAR LAYER
When 0 is fixed, (0,))) reaches its minimum value when:
Theorem 4.5 shows the conversion of linear layers under differential coding in ANN-to-SNA conversion.
[25-110
Eim1 ers
2n
1/2r
Theorem 4.5. For linear layers, including fully connected and convolutional layers that can be represented by Equa- tion (22).
== k1
125-1)0
1 Σ
2: erf
2m
V20
(27)
x:' W--''='++''
(22)
(21-110
2n
ynn 1e
where W and b' is the weight and bias of layer I. Under differential coding in SNNs, this is equivalent to eliminating the bias term b' and initializing the membrane potential of the subsequent layer with the bias value.
0
V
(21-110 110
1 >
2-_1 R
erf
2n
v/2
The proof of Theorem 4.5 is detailed in the Appendix F. Figure 3 shows the overall method to replace ANN modules by SNN modules under differential coding.
Lemma 4.9.
QFa k.) = 1 (f2(2,0,)) max (2r, 01)²
(
dn, (28)
f2(2,0,k) &elamp 1.0,N
(29)
4.2. Optimal Threshold for ReLU
When is fixed, (2, reaches its minimum value when ==
When replacing the ReLU function in CNNs with spiking neurons, we propose an algorithm called threshold iteration method for determining the optimal threshold.
Algorithm 1 Threshold iteration method to find the best
Assumption 4.6. According to (de G. Matthews et al., 2018), 8), assume that the input x: to the neuron follows a nor- mal distribution X with mean and variance o².
threshold
1: Input: Pre-trained ANN Model FANN (W), Dataset D.
2: Initialize: Set <- 1 (any positive initial value)
3: Run the model FANN (W) on dataset to statically compute the mean μ and variance σ² of pre-activations of each ReLU separately.
Based on Assumption 4.6, we introduce Definition 4.7 to define the overall error function, which is obtained by inte- grating the function error over the distribution of activation values.
4: repeat
5: Update k based on la and o² according to Eq (27)
Definition 4.7. In the T' time-steps conversion, the quan- tization and clipping errors of the ReLU function can be expressed as
6: Update 0 + k1 0
7: until <<<<<< 4 < + c, where tends to 0.
Output: Threshold 0
QE(O)- - (f(r, max (x, 0))²
(m-w² 20
dæ, (23)
f(x, 0) 9. clamp (1
(I", [M J.
According to Lemma 4.8 and Lemma 4.9, we obtain the following inequality and Theorem 4.10:
(24)
where [xx, 0) represents the expected encoded activation in SNNs for a threshold 0 which is proposed by (Bu et al., 2022a). For an IF neuron, N T. For a multi-threshold with n threshold, roughly let T"..
QE(k,0) < QEg(k10, ( QE, (1) < QE(O).
(30)
Theorem 4.10. Starting from any positive initial value of B. the rate of change ky can be continuously calculated based on the prior mean μ., variance o², and the current threshold using Equation (27). The ileration convergence, which point the global optimal threshold O is obtained. The process is guaranteed to converge long as the threshold is greater than 0.
ki continues until
Finding the optimal threshold by directly differentiating this function is challenging. However, we can take an alternative approach by introducing a variable & to help determine the optimal threshold. We consider two cases: k multiplies the output threshold amplitude in Equation (25), and k multiplies the threshold during spike calculation as in Equation (28). These cases yield the following two lemmas. Lemma 4.8.
The proof of Theorem 4.8, 4.9, and 4. are detailed in the Appendix G. H and I. Therefore, the optimal can be determined by the Theorem 4.10 and Algorithm 1.
QE, (0, k) 1_ (f1 max 0 - dr, (25)
4.3. Hardware implementation of MT Neuron
fi(a, O, k) = ks clamp (1 NN= (1 1,0,N
Equation (6) in Section 3.1 is presented for ease of under- standing. In hardware implementation, the argmin module
(26)
6

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Differential Coding for Training-Free ANN-to-SNN Conversion
is not used. We develop a hareware-friendly version of the MT neuron model, which efficiently map the appropri- ate threshold using the potential's sign bit and exponent bits at an extremely low cost.
i < n, S = 0 and i == 1 E,
1, if
MT (md
i 12n, n, S = 1 and i - == -E, 0, otherwise.
Compared with previous ANN2SNN methods, the MT neu- ron is required to transmit an extra index for the threshold. When implementing the MT neuron, two implementations can be considered:
(34)
For differential neurons, the memory overhead compared to initial neurons, such as IF or MT neurons, only includes an additional membrane potential. This extra potential is used to adjust the input current as described in Theorem 4.4.
1. Sent [i S[(] to the next layer
2. Add an external threshold dimension with 2n elements to STt). set [| = 1 and S[t][]] = 0 for all j * i. At the same time, an external threshold dimension is added to the weight of the next layer, whose elements are the multi-level
To enable fast execution on GPU, we also design an efficient algorithm which is detailed in Appendix P.
thresholds.
5. Experimental Results
For simplicity, use implementation 1 on GPUs, which is not pure binary but equivalent to implementation 2 with binary outputs. The MT neuron is also compatible with asynchronous computing neuromorphic chips because its outputs are still sparse events. Take the speck chip (Yao et al., 2024) as an example. The LIF neuron in the con- volutional layer in speck chip outputs (c, xr, y) to the next layer. When using the MT neuron, the only modification is adding threshold index, i.e., (c, x. U, i). The computations of the next layer should also be changed by using bit-shift operation on the weights, as the threshold is a power of 2 and this allows multiplication to be avoided. After the above modifications, the computation is still asynchronous and event-driven. The implementation to avoid argmin in Equation (6) in hardware can be described in the following two steps.
In this section, we first evaluate the performance of our pro- posed method on ImageNet dataset across different models, comparing our results with state-of-the-art ANN-1o-SNN conversion methods. Then, compute and analyze the energy consumption of the converted SNNs. Finally, we con- duct comparative experiments to validate the effectiveness of differential coding and the threshold iteration method.
Table 1. Accuracy and energy ratio of DCGS(Ours) of different converted models ImageNet Dataset
Model Config
Time-step T
Acc/Energy
2
4
8
12
16
ResNet34-4/1, Param:21.88 1.8, Acc:76.42%
Acc
59.71
73.35
76.04
76.26
76.35
Energy ratio
0.14
0.24
0.37
0,46
0.53
VGG16-4/1, Param:138M,
Acc:73.25%
Step I: Get SNN weights by using the weight normalization strategy (Rueckauer et al., 2017) described by the following equation.
Acc
70.69
72.72
73.17
73.23
73.26
Energy ratio
0.10
0.15
0.22
0.26
0.29
ViT-Small
Param:22. IM, Acc:8
1..8%%
0¹ BI+I*
Acc
77.84
81.11
81.43
81.39
81.38
1/ BLANN W. W. SNN ANN
(31)
Energy ratio
0.32
0.62
1.05
1.39
1.71
6
NNNN-
(32)
5.1. Comparison with the State-of-the-art ANN-to-SNN Conversion Methods
gi+1*
We then set all base thresholds 0¹ = 1, resulting in the following thresholds for the MT neuron:
We conducted conversion experiments on 11 different CNNs and Transformers using the Imagenet dataset. We denote the converted model as model - n/c. where the multi-threshold neurons have n positive and n opposing negative thresholds, and the calculated channel-wise thresholds are scaled by a factor c. Eg., the ResNet34-4/2 model represents the conver- sion using the ResNet34 model, employing multi- thlti-thrrehold spiking neurons with 4 positive and 4 negative thresholds, and the actual thresholds are based on the statistical thresh- olds multiplied by a factor of 2.
X
<<i S n,
(33)
2 <i < 2n.
Step 2: We define fm²ld [1] = (- (-1) M) with 1 sign bit (S), 8 exponent bits (E), and 23 mantissa bits (M). Since the median of Σ and 2 is 1 we can easily select the correct threshold index i using E and S of without performing 2n subtractions to calculate the argmin in Equation (6):
4 3 mm((t)
When n = 1, it can be treated as an IF neuron with an additional negative threshold. Table 2 shows a comparison
7

---

Differential Coding for Training-Free ANN-to-SNN Conversion
Table 2. Comparison between the proposed method and previous ANN-to-SNN conversion works ImageNet dataset.
Method
Type
Arch.
Param.(M)
ANN Acc(%)
T
SNN Acc($))
TS
CNN-10-SNN
VGG-16
138
72.40
64
70.97
(Deng & Gu, 2021)
SNM
CNN-to-SNN
VGG-16
138
73.18
64
71.50
(Wang al., 2022a)
MMSE
ResNet-34
21.8
75.66
64
71.12
CNN-1o-SNN
(Li et al., 2021)
VGG-16
138
75.36
64
70.69
QCFS
ResNet-34
21.8
74.32
64
72.35
CNN-to-SNN
(Bu al., 2022b)
VGG-16
138
74.29
64
72.85
SRP
ResNet-34
21.8
74.32
4, 64
66.71, 68.61
CNN-to-SNN
(Hao et al., 2023a)
VGG-16
138
74.29
4, 64
66.47, 69.43
MST
Transformer-o-SNNY
Swin-T(BN)
28.5
80.51
128. 512
77.88, 78.51
(Wang et al., 2023)
STA (Jiang al., 2024)
Transformerto-SNNN
VIT-B/32
86
83.60
32, 256
78.72, 82.79
SViT-S-32I.evel
22.05
81.59
64
81.45
SpikeZIP-TF (kang you et al., 2024)
Transformerto-SNNN
SViT-B-32Level
86.57
82.83
64
82.71
SViT-L-32Leeel
304.33
83.86
64
83.82
ECMT (Huang al., 2024)
ViT-S/16
22
78.04
8, 10
76.03, 77.07
Transformer-to-SNe
EVA-G
1074
89.62
4.8
88.60, 89.40
ResNet18-1//
11.7
71.49
32. 64
69.89, 71,08
ResNet34-1/1
21.8
76.42
32. 64
58.86. 74.11
VGG-1/1
138
73.25
32, 64
72.04, 73.13
CNN-to-SNN
ResNet18-4/1
11.7
71.49
4,8
70.07, 71.31
ResNet34-4/1
21.8
76.42
4, 8
73.35, 76.04
VGG-4/1
138
73.25
4, 8
72.72, 73.17
DCGS(Ours)
VIT-S-8/4
22.1
81.38
2,
77.84, 81.11
ViT-B-8/4
86.6
84.54
2,
80.34, 83.98
ViT-I.-8/4
304.3
85.84
2,4
83.73, 85.45
Transformerto-SNNi
EVA02-T-8/4
5.8
80.63
2,4
66.32, 79.56
EVA02-S-844
22.1
85.73
2.4
71.37, 84.70
EVA02-B- B-402-B-8/4
87.1
88,69
2.4
84.62, 88.16
EVA02-L-8/4
305.1
90.05
2.4
88.25, 89.72
of our method with other ANN-to-SNN conversion methods, and detailed results can be found in Appendix K.
relative the ANN, with EMAC = 4.6pJ and EAC = 0.9pJ.
EsNN EANN
M ACssNN * EMAC + ACSSNN EAC
In CNNs, when 1, our method outperforms the existing methods on the same structure achieving state-of-the-an results; and when n > 1, we achieve better performance with extremely shorter time-steps.
MACSANN EMAC
(35)
Since most computations in the network oceur in the fully connected, convolutional, and matrix multiplication lay- ers, which in SNNs are primarily implemented by addi- tions (with ACssNN >> M AC'ssNN) we approximate M ACssNN R O. We then use the statistical spike emission ACsss rate to estimate M ACCsss thereby estimating the energy consumption of the SNN relative to pre-conversion ANN. Table presents partial results, and the detailed results for all converted SNN models can be found in Appendix K.
In Transformers, the threshold iteration method is not suit- able, and using the top 99.9 99.9% of activation values does not optimal thresholds. As result, achieving high performance with n in short time-steps is challenging. Therefore, we scale the statistical thresholds by 4 and setting = 8. Our method requires no training and achieves high performance in extremely short time-steps.
For CNNs, our method achieves SNN performance compara- ble to the ANN with low power consumption and extremely short time-steps. Notably, for the VGG16 model, it achieves an accuracy of 73.17% with only a 0.08% accuracy loss and 22% power consumption.
5.2. Energy Estimation and Result Analysis
Based on (Horowitz, 2014), we use Equation (35) to esti- mate the energy consumption ratio of the converted SNN
8

---

Differential Coding for Training-Free ANN-to-SNN Conversion
10
20
Apcnzoo (96)
Rtatio
Eoegy Ratio
Accurccy 2
0.
003
0.5
30
00
30
Time-stpp T
Timo-step T
RenNet33-4/2, DC Ace
ResNet34-4/2. 4/2 DC Energy
ResNet34-1/9 1/2, TI Ace
ResNet34-1/2. 1/2, TI Energy
4/22 RC Ace
ResNet3 4-4/2, 4/22, RC Energy
ResNet34-1 1/2, LA Ace
ResNetD4-1/2 LA Energy
VIT 81a-8-aa1-8/44 8/4, DC Ace
PTTT-Smal-8/4 8/4, DC Energy
R/s
ResNet3444/2; Energy
ViT-Small-8
Sir--mall-84 Energy
ResNet3t-422, LA Acc
RenNet3t-/22, LA Evergy
Figure 4. Effective of the differential coding compared the rate coding. DC and RC represent Differential Coding and Rate Coding, respectively.
Figure 5. Effectiveness threshold iteration method. TI LA represent Threshold Iteration Method and 99.9% large activa- tion method respectively.
For Transformers, although our method achieves high accu- racy with extremely short time-steps and shows decreasing energy consumption growth rate, there is still significant room for further optimization. This is primarily due to the lack of an optimal threshold calculation method, which causes inefficient spike firing in the SNNs. This leads to larger errors when matching the ANN activation values, sulting in more premature spike emissions. This is area we aim to improve in future research.
the top 99.9% of activation method, with different num- bers of threshold neurons in ResNet34, Here we set scale factor e = 2 to prevent the accuracy from being too small when using the 99.9% large activation method. The partial visualization results are presented in Figure and more information can be found in Appendix M. The experimental results show that the thresholds derived using the thresh- old iteration method outperform those obtained through the 99. 9% large activation method, achieving better accuracy and lower energy consumption at each time-step.
5.3. Effectiveness of the Differential Coding
6. Conclusion
To validate the effectiveness of the differential coding, we compared the performance of differential coding and rate coding using the same model. The partial visualization re- sults are presented in Figure 4, with a more detailed table provided in Appendix L. The model using differential cod- ing not only outperforms the rate coding model in terms of accuracy, but also consumes less energy. This because dif- ferential coding directly updates the current encoding value based on previous results, avoiding decay. It can represent broader range and steadily improve representation accuracy. Once the representation precision reaches certain level, no further spikes are emitted.
This article introduces training-free ANN-to-SNN con- version method based on differential coding, Instead of directly encoding rate information, it uses spikes to encode differential information, improving both network accuracy and energy efficiency. For ReLU conversions, it includes threshold iteration method to find the optimal thresholds, which further enhances the network performance.
However, the proposed method also has some limitations. Differential coding requires spiking neurons to have at least one negative threshold to generate negative spikes for er- ror correction; otherwise, excessive spike errors will ac- cumulate continuously. Meanwhile, we have not develop method to determine the optimal thresholds for Trans- formers, which limits the conversion performance on Trans- formers. Future research could focus on addressing this challenge.
5.4. Effeetivvnesss of Threshold Iteration Method
To verify the effectiveness of the threshold iteration method, we compared the performance of the converted SNNs us- ing two different methods, threshold iteration method and

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
Acknowledgments
Cordone, L., Miramond, B., and Thierion, P. Object detection with spiking neural networks on automotive event data. In 2022 International Joint Conference on Neural Networks (LJCNN), pp. 1-8, 2022. doi: 10.1 (009UUCNNN50044222229999999
This work was supported by STI 2030-Major Projects 20212D0200300, the National Natural Science Foundation of China (62422601, U24B20140, and 62088102), Beijing Municipal Science and Technology Pro- gram (2241100002244004) Beijing Nova Program (20230484362, 20240484703), and National Key Labora- tory for Multimedia Information Processing.
Davies, M., Srinivasa, N., Lin, T.-H., Chinya, G., Cao, Y., Choday, S. H., Dimou, G., Joshi, P., Imam, N., Jain, S., Liao, Y., Lin, C.-K., Lines, A., Liu, R., Mathaikutty, D., McCoy, S., Paul, A., Tse, J., Venkataramanan, G., Weng, Y.-H., Wild, A., Yang, Y., and Wang. H. Loihi: a neuromorphic manycore processor with on-chip learning. IEEE Micro, 38(1):82-99, 2018. doi: 10.1 10.1109/ 112130359.
Impact Statement
This paper presents work whose goal advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
de G. Matthews, A. G., Hron, J., Rowland, M., Turner, R. E., and Ghahramani, Z. Gaussian process behaviour in wide deep neural networks. In International Conference on Learning Representations, 2018
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Differential Coding for Training-Free ANN-to-SNN Conversion
A. Overall Algorithm
Algorithm outlines the whole procedures we adopt.
Algorithm 2 Differential Coding with Graded Units and Spiking Neurons (DCGS) Conversion Method
Input: Pre-trained ANN model FANN W), Dataset D. Time-step T, or Threshold percentage p and scaling factor c.
2: Output: Converted SNN model FsNN (W, v)
3: Step I: Determine the Threshold:
4: if FANN(W) is ReLU network then
5: Use the threshold iteration method with 7' to calculate threshold on dataset D
6: else
7: Static threshold 0 as the top p% of activation values on dataset D, and multiply by the scaling factor e
8: end if
9: Step 2: Replace Modules:
10: Replace the nonlinear layer with differrnniial graded unit.
11: Insert differential identity spiking neuron before each linear layer.
12: Remove the bias b from the linear layer and set the initial potential v = b for the next layer.
13: Return the converted SNN model FSNN (W,6, v)
B. Explanation of Definition 4.1
Definition B.1. (Repeated from Definition 4.1) In differential coding, the encoded activation value r' is defined as shown in Equation (14), where eL( represents the encoded output value of the neuron at time-step I, and a represents the actual output value of the neuron. The relationship between the two is expressed by Equation (13), as follows:
[1] r'( + xx [1]
(36)
t
[t] = r'tt 1] + t
e'(
(37)
where starts from 1, r0] =
Proof. In this definition, e is essentially adjusted based on the historical encoded values. no spike is emitted, then e'[1] [t 1]. also ensuring that the encoded value r'[ r''(t 1]. The derivation of Equation (37) be written as:
x [t]
r r'd = r'
1]
r'lt
x [t]
t
r
e'(t]
r' - 1]
e/[t]
etit 1]
t 2
(38)
+
11
13

---

Differential Coding for Training-Free ANN-to-SNN Conversion
C. Proof of Theorem 4.2
Theorem C.1. (Repeated from Theorem 4.2) Let F be a nonlinear layer with only one input x 10), such as Gelu, Silu, Maxpool, LayerNorm, or Softmax. In LANN-to-SNY conversion, the mapping from F dynamics of the differential graded unit differential coding is given by Equations (15) and (16).
m [t] [t] m'lt f + x0--1(1)
(39)
x' [4] = + (F (m (4) - F' F'mmmm' (m (0/--1]. 1)))
(40)
where m is the membrane potential time-step which is equal the encoded input value, r is the encoded output activation value of the previous t time-steps. The output of layer I time-step t, which serves the input to layer I + 1, is given by xe [t).
Proof.
m = (4 = - 1] + x² x'1 m²(l
+
(41)
t
x' = (t r' 1]) ** (F - [t) F(r'-I(t
It 11)) = t= (F(m' F(m'(t 1))
(42)
From Theorem 4.2, a single-input unit requires two variables: one record m [t] and another record F(m'(t)) in order to reduce redundant calculations at each time-step.
D. Proof of Theorem 4.3
Theorem D.1. (Repeated from Theorem 4.3) Let be an operation with two inputs, such as matrix multiplication or element-wise multiplication. ANN-to-SNN conversion, the mapping from operation to dynamics of the differential graded unit differential coding is given by Equations (43) to (45).
x. A [1]
m 1 [t] = r [t] = m - 1]
(43)
1- [t]
m a [t) -r
[-1(t] [t] m'ylt 1]
B
(44)
#'(t] - [t] o + . m [t] + m [t)] x (t).
(45)
where m'Al4 and m'g(4) are membrane potential at time-step and r [1] and r (1] are the encoded activation values of the previous layers at time-step t. The output of layer at time-step which serves as the input to layer I + 1, is given by a! (')).
Proof.
1
m - 1) + a (4
m = - [1)
1] +
(46)
t
L [-1(t]
m/glt 1] + (
m/ilt = r [t] = / 1- it - 1] +
(47)
x! [1] 1 (r' [t] - r' 10-1) 1))
(m m [t]
m 2
1]
mly 1])
= * (mmaa lt - 1] +
x
1-1 [t]
A
(m/
- 1] + a/3 [t]
-m 1] m 1))
(48)
1- [t) 2//3"
a [||]
m
1] + m' (1- - 1] x B (4)
t
14

---

Differential Coding for Training-Free ANN-to-SNN Conversion
From Theorem 4.3, a neuron with two inputs requires variables record m'll and m/ respectively.
E. Proof of Theorem 4.4
Theorem E.1. (Repeated from Theorem 4.4) In rate coding, the output of the previous layer, [t), is directly used as the input current for the current layer [t] = xt_ In differential coding, the input current [t]] can be adjusted as shown in Equation (49), which converts any spiking neuron into a differential spiking neuron:
F'4- = =m + x." [4].
(49)
m 1] ml[(] m] + x² x [
(50)
6
where m0 is b²-2 if the previous layer has bias else O.
Proof. Let the expected input encoding value of the differential neuron at the I-th layer at time-step / be r/] and the expected output encoding value be r' [t]. Due to soft resetting, the total expected membrane potential change is m4, [t] rl r'(t - 1]. The total input current is then:
r't rf 1] r' 1 x [t] m [t] + (t
(51)
Since by Definition 4.1:
x
[t]
+
(52)
t
x. [t)
[t] i
1]
(53)
have:
mf, [t + 1] rd12((t] rr((t)
(54)
_d- it)
x
-
1 +
1]
(55)
t
- [1]
ao [4]
=m
(56)
F. Proof of Theorem 4.5
Theorem F.1. (Repeated from Theorem 4.5) For linear layers, including linear and convolutional layers that can be represented by Equation (57),
a' = W d' +
(57)
where W and b' is the weight and bias of layer
Under differential coding, this is equivalent to eliminating the bias term b' by initializing the membrane potential of the subsequent layer with the bias value or adding the bias to the first input current of that layer, and then running the dynamic process at each time-step according to the following equation:
x
[t]
W
(58)
Proof. In the context of ANN-to-SNN conversion using rate coding, the output x'tt] of layer can be expressed as:
x [
W!
+b
(59)
15

---

Differential Coding for Training-Free Free ANN-to-SNN Conversion
Under differential coding defined in Definition B.I, we have:
e [t] = r'It - 1] + x: [
(60)
r [4] = r'( - 1] + x [t]
(61)
r'10]
(62)
e'(1] = W er(e'_(0) + b'
(63)
[[] /' + +bb
(64)
For > 1, the following transformation holds:
x' [t] = et() - r' 1] = + ('] (t - 1) * + i 1] - r
b¹) + 1 (1] b' w'
(65)
= * W/ - [1]
when / = 1, we can let [0] = b', and initialize the membrane potential of the subsequent layer with the bias b' or adding the bias to the first input current of that layer and start running from time-step 1:
x' [t] e [t] - r'It 1] -
= w (1) + b'- w''=w'''-pp W w'
(66)
Therefore, for any t> 0 in differential coding, we have:
x' [t] = W x 1'1
(67)
G. Proof of Lemma 4.8
We first prove a previous lemma before proving Lemma 4.8. Lemma G.I.
f (c x)²ee= ( dr = 30 / (o² 2
2cp 2) erf (b- 11) ) erf
(a 11
/20
20
(68)
16 w) ws?
(u 2)
+ (-o2 (b + -2e)
20 + (0² (a + 1r 2c))
20
f e جرري
dr, fo De dr and So we (==
(m=e)2
Proof. We first calculate
dr separately.
Since (x) =
VF Jo -t Jt 1oº e -
2
l/ erf (x)
=
2
(s=p)²
dr V20 {" c
(m-p)2
20
f (xr ((--)) 11)
V20 v
erf
(b (bb-w) erf (1a ((a-j)
V2a
V20
22
(69)
= = go erf (b- p1)
(a- 14)
erf
V20
V20
re 20 dr = 1 (xr - μ 14)
(m-p)²
(e-w)²
20 dr 0 S
bb (m=w)²
(x (x- n12
(a-n)²
20
11 e
dr
20²
a
(70)
-2 - (0-u²²
- uu 3 (
(a-w)²
(b (1)
(a 11)
ctf
erf
22o
20
16

---

Differential Coding for Training-Free ANN-10-SNN Conversion
x²e 20 dr ! ( 11²² + 22 (ar (1) 1²)
(x-n)
(x-ni²
20 dx
=0- S0² (x - 1)) de
+ S0 (2 (x - (4) + ++)) e
dx
la-w)² 20
(m- 20
-002 -22( (b- 11)
20
+ 0² (a- -11)
+ 2
d (x - 1i)
1b-p)2
la-ple
(b (b-11)
(a2 (a-11)
V2a ))
11)
20 11 c
20
20d
+ 11
20 erf
- erf
(71)
V2a
(6 2)
la 41²
(b 14)
((a-1)) 11)
(a
= (b- 12)
20
+ +o² (a- -ute
20
+
20
erf
- erf
V2o
220
20²21 (o
10-p)²
(a-mpe
11 Vi0 20 erf
(b (4-
(a (n-11)
)
11)
20
20d
- erf
V2a
22a
@b e2² 20
iu 1 + 2 (o² + x²2)
(b {b- (4)
(la-1) 11)
(-o2 e
+ (a²
erf
erf
20
22
Finally, aggregate and calculate the answer:
(c- z)²²
(m-p)² 20 dr = I- (x² -2er 2cx (7) e-
(m-p)²
20 dr
(h-p)² 25
(m-p)²
erf (b- (b- (4)
((u-w)
(-0²
(b+ 1))
+ (o² (a m)) e 20
+ V 20 (
erf
V20
1220
(m-w) 20
+ #HV N0 erf (b (b-p) p1)
(a - n1)
2c
-02
e- 20-
20 -
erf
22
/2a
(72)
+ 2 0 erf (b (b-w) (4)
(a- 11)
erf
V2o
V2a
(b - 11)
(a 11)
0 (o² + 1 -2c4 + )
erf
erf
V2o
V20
+ 0-02 (b + 2c)) e m0-m1²
(n-w)²
+ (o²
(a + μ 20))
17

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
Then, can prove Lemma 4.8 base on Lemma G.I.
Lemma G.2. (Repeated from Lemma 4.8)
(E,(0.K) = - +00
01(2,0,k) max (z, (),00) 203 dx
(73)
(1=a [ $.0000
Nx +
0,), N
celmp
(74)
When 0 is fixed, Q0,(0, k) reaches its minimum value when:
(2i-
1 Ei=1 er
2n
(26-1)0
$2a
2n
0
1 =
k: k1
(75)
(2i-
V
(2i-1)0
1 Dx=1 Lr-f
2n
1
"
n erf
/2a
i=1
v2a
Proof. According to Lemma G.I, have:
S (c (c-x)) 2)2c (c-2)²
-2e + c7) erf (0r((1/)) ((b-w)) - erf 1m- de - - (02 (02 +
(a-1)) 11)
(a
V2a
22
(76)
1 w²
+ +--0² (b 1 -2c)
20t
(0" (a + - 2c))
20
Expand the calculation of QE::
max (2,, 0) )
nar 2
( dr
k clamp
J,0,n
n
8
nx +
(
k- clamp
J.
0, n
2o dx
0
(x)²
+
n-1
-
10
(emui
dr
+00
(k0 =)²e
(--w)
dr
(2n-1)9 1)0
(-1
6
-
20
(02 + +122)
erf
0
2n
220
14)
erf
V2o
=
+ (-0² (20 + +x))
2m
202
+ (a²N) e 20
(77)
(21++1) 2n
(2i-1)0
2
0- + 3
2k
i0
1
(1: hm
10
2
erf
11)
erf
2n
V2o
V2o
(2i + 1)0
10
(104 =)² 2o2
a2 ( (2i - 1)
10
4n
2k
+ ft
2h
2+2
2n
n
2n
n
(2n-1) 2p
(2n-110
o
[22 + 112
2k0 (10)2)
1-erf
22a
11
+
(g2
(2n-1)8
2m
14 22:0)
2n
267
18

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
(2/ -1)
m
o (o² + +2²
ler
erf
-11
+ (-0²
(cn + 1)) e
2n2
+ +
e 202
V2o
<200
(2i-1)8 2n
(i - 1)0
(i - 1)
11)
0² + 1² 2k
Ve + 222
erf
2
n
V20
(2i-1)0
0
a² + 1
2k
i0
1
+ k2
i0
erf
2n
n1
220
-02
(2i 1)0
+ P 2kc
(i 1)
[26 118
2m
20
n-1
0²
(2i 1)0
+ f
-2k
i0
2n
20
112 -,.)
2n
n
2n
12
1=2
i=1
(2n 170
(2n-1)0
+
20
(02 + +1² - 2k-0, + (k0))*
1 -
-re erf
2n
H1
+ 2
(2n 1)0
+ 2k0) )
2
V20
2n
2n - 11)
V o0 (o² + +²² lrf
erf
-H4
+ (-0²
( + m)) e
202
+ ((²pp) e 200
V20
V2o/
[2n-1)0 11)
1 2
0² + 112 2k
(n- 1)0
/
+ 122
(n
1)0 (0-11)0)
2
erf
2n
n
20
( 2
0
11)
0
0² + 11² 2k- 1 + 2
erf
2m
n
n2
V2a
+ (00
(2n- - 1)0 0
+ k - 2k
(n- 1)0 0
e
(2 1)
2n
2,2
+
02
+ μ - 2k .
2)
e
(2n -u)²
2n
n
2n
11
(2n 13
(2n-
V 30 (02 + 1²² 2k-0p + (10)))
1- --erf erf
2n
H1
(02
(2n-1)6
+ 1 2k0)
2n
202
22
2n
0² + 22 2k (i 1)0
u + 2
(i 1)
2
2
-12 + 2h
i0
2
i0
2
erl
(2i-1)8
2n
$1
n
n
n2
/2a
(2i 1)
1)8
(2i 1)
i0
126
112
2
+ p
- 2h
+ 0²
+ p - 2k
e
2
2n
n
2n
n2
J (0² 2 + +12) (1 (1-erf
erf
(T
V2o ) + (o² (1)
e
20?
+ 00
(2200p + (k0))²
+
2h |
0
k²0²
2n - 1
erf
(2n-1)0
2n
11)
+
2
2k:
0
1
1.2
0
erl
(on 11)
n
V20
n1
V22
(2n 13e -p) 2
+
2
2k-
0
2n
202
+ 2
2k
0
202
T1
n2
120-1)0
2k-
0
1.2
0
2
(2i 1)
erl
2n
11
71.
//o
σ 2k 0
(38 118
-1)
e
1l)
19

---

Differential Coding for Training-Free ANN-10-SNN Conversion
20 (o² + 1²) 11 - 11-rff erf
) + +o²2
20?
V 20
-2k0pu (k0)²)
V20
0
(0)
0 2
[2i-1]6
11)
2k p k2
(2i 1)
erf
11
22a
0
2k
20
(2i-1)0
2
1-2
V V5000
V
2
(2i 1) erf
2n
220
1=1
(2i-1)0 2n
14)
(2i-110 130
+ k
2 1 0 Σ erf
2/ 200pp - 20
2m
20
V20
1=1
1=1
202
(o² + x²²
(1-ort
- (1)))
+ (d²1) E
/20
Since this is quadratic function with respect to k;, the value of that minimizes is:
126-110 110
2 20700
E fer
Zn
-1
20020 Ei=1
[2 110 -p) v2a
2 F002
1
Ei=1
n= erf
(78)
1 1 Σ erf
(11--110 14)
(2t-110
V2o
Σ Le
202
(21-1)0 14) v/a
12c-130 p))
1 1- ( erf
1- >n
( erf
v/20
H. Proof of Lemma 4.9 Lemma H.1. (Repeated from Lemma 4.9)
(E2 = 1 +oo
(m-w) 20 dx
(2((,,0kk)
max (2r, 0))² e-
(79)
0
ke Nx k0
f,(2,0K)) N
clamp
3 11,0, J.O.N
(80)
When 0 is fixed, QE2(0, k) reaches its minimum value when 1.
Proof. Expand the calculation of Fg:
+00
nI + kd 2
2 max (x. 0)
(
clamp
1,0,n
20
dr
k9
(81)
Ak Zn
(2i+1100k
(
dx
(2i 1)0k
21
i0
x
2
(m-n)²
dx +
(2n 12n-1)0k 1)0k
+00
(0 =))²
20
dr
2n
2n
20

---

Differential Coding for Training-Frre Frrennngg-rreeeee ANN-to-SNN Conversion
00
(o²
+1² +
erf
2n
m1
erf
+
(-0²
(BL
+
+n]]
20
20a
(O²N)
277
20
<20
10
1)
[2i
1j0k
1)
02
2
orf
erf
72
V2a
v 22
(2i- 18tk
Σ -02
2m
02
(2i- 1i6k
10
Zn
2m
1
2m
+ j
2
n
20rd
1=1
(2n-1)0k
(2ts- 136k
V 20 (0² + 11² 20p + 02)
orf
2n
+
(o²
[22m-1100 110k
+ 1 20
2m
V20
2m
0h
(ot -M]
2xc
V fo (o² + 1122) crl
erf
+
(-o²2
(27
p))
20
(o²N))
$20
220
(2n-1
(2n-1)0k
V (02
+ 112 20 + 19)
--ef
(02
(22-1108
+ 1 -20) 0c
2022
V2a
0² + 12²
(i
1)0
(i
1)00
2
erf
110k
21
a
20
(21 1,0h
(1)
2
2
10
10
orf
2m
/2
12
(2i- 118k
-02
12:11 2m
(1
1)6)
207
1-1
(2i-118k An
2
i8
207
12
mi)
Vio (d2 1n²2)
crl
orf
(
d²
(20
n))
(0" u))
22a
VZo
(2m-110k
110k
V 10 (02 + +1²²- 120p + 09)
cef
(22
(2n 110k 2m
H 20)) 0c
22r
5/50 30
0² + 21a 2(11 1)0 p + (n
1)0)
erf
120-1 2n
11)
Via
0² + 112
0
+ (2))
eef
H)
n
V20
r
20
-2 (i 170 +
(i - 1)8
10
i8
12i- 2n
1
V o0
exf
n
v20
1m2
(2t-1)99
-2
(2n 1i8k 2n
p 2
(n 110
7
2.2
Ak
2n
Y2
0
2n
20.2
(26- 1ook
n-1
0
12-1)00 1jok Zm
26 1)00
7
+ +
(21 (21-130k
Zm
2
10
72
207
(0²
-err -H ((²N)
10 + v f0 (0² 11² 28p 88))
/2a)
(
(2n-1)0k
(0) 2 (n)
p))
2 1 +
(-2i + 1)
erf
v o
2
erf
V20
v/20
(2-11
2 8
8
11)
(-2i + 1) erf
1
n
220
1)ok 1)9k
200 8
2n1
02 (22" 9
11)
n2
(21-1:00 1:00 2m1
0
20
n2
21

---

Differential Coding for Training-Free ANN-10-SNN Conversion
20 o (o² + +1²
127) (1 (1-erl ere (Tm
#² + (o"N))
E
20
+
+/330 (-20p f + +
V20
(2i-1)6k
0
0
(2i-1)0k 2n
11)
n
2n
0
2
2
(-2i 2i +1)
erf
-20
e
11
72
V20
11
o (o² nr) (1 (1-erf
erf
-M6
(o²u)
20
V 1
(-20p 02)
/20
(2i-1)0k
(2i-1)0k
(2i-1)0k
0 + 20 -
V
uerf
2n
11
+
1
2i
erf
2n
11
oe
2x
-H
22
120
1=1
We only need to extract the part containing k, denoted L, for calculation.
[2i-1]06 110k
(2i-1j0k 2n
22 110k
1)0
11)
L
V t erf
erf
(82)
202
2n
/20
Since erf (x) =
2 =2 V
ee
(2)
139k
1988
E
Σ
axd
110
22n
2n
V
(2)
1)8
2
(12- 150
V20
20
de
2r
14
[21
13
120-110k
(2t-110
(24 1)
13-1
12--100
2n
(83)
(121-1)96 199k In
(2hm 1100
11-11
(2V 1)
(21-1)0k
(21-110 201
2n
2m
12
1)²
1)e
So, the value of that minimizes the function is k 1.
Proof of Theorem 4.10
Theorem I.1. (Repeated from Theorem 4.10) Starting from any positive initial value of 0, the rate of change ky can be continuously calculated based on the prior mean variance o², and the current threshold 0 using Equation (27). The iteration = k continues until convergence, at which point the global optimal threshold is obtained. The process is guaranteed to converge as long as the threshold is greater than O.
Proof. Assume the optimal threshold is 0, then according to Lemma 4.8, after one update ko should be equal to that 1
To prove that 0 is the optimal value, it is equivalent to proving that the following equation has unique solution:
(2i 1)8
(21-1)8
Lerf
1=
</a
2n
-P
Σ Ie-
1
n
=1
(84)
0
(2i-1)8
(2í-1)8
1 - Di=1
2i-1 erf
2n
V2a
-1=1
2i-1 2i-lerf erf
2n
equivalent proving that f (0) has an unique root.
22

---

Differential Coding for Training-Frie Freeingg-Freeeee ANN-to-SNN Conversion
(2i-1)0 v2a
(2- 1:0
(2i-1]0 -p)
(0) = " |- -
terl
-p)
11
2n
-
--E=1
2i
V2e
=0
(85)
Step 1: Calculate the first derivative of f
Since erf' (x)
V
(25- 13
(21-1))
(0)
24
24
(2i 1)
2n
(2) 110
2m
11
(2i -
1)
VT
2n
2r²
2x
(2t-19 110
(2-1119
24
(2i 1)
-1
erf
0
1/2
V20
n12
VT
2re
(86)
V
2
-/1-
+
2i
(2h-116 118
2n
(2i - 1)
2ny/2 22
(26- 1ie
Zn
Σ
2i
n2
erf
(2i-1)8
2n
V20
11)
(26-116
(2v-1)0
(2)
orf
22r
2n²²
2
Step 2: Calculate the second derivative of? f
(21-110
2m
(2i-110
1"
(0)
TT
2
Σ
21
n2
v
(2i 1)
2nv2o
VT 1=1
2
2i
2n²²
(2) 1)
2n 2n
2m
26a
1=1
8
2í
(2i - 1)
(2i-19) 1i
2m
-
2n
136
(2i - 1)
(87)
VT
n²
20122
2
(26-110
12:-1W
2/
1)
())
(2) 1)
3 0
nd
2ne
r²
2n
Step 3: Analyze the trend of the second derivative (0)
To analyze the changing trend of f' (x) on the interval from to positive infinity, we start by examining the given expressions:
3 n
2i - (vr))²
2i
("(0)=
(88)
V
n²
2n
and
(= =
(89)
[2 118
2
2m vBe
The behavior (") as 0 increases from to +oo depends on two parts: e
and
((2i-1)6
0 #) Since
0, the sign of f") is determined by the second part. This term
decreases from 3 to negative infinity as 0 increases.
Thus, ("(() initially decreases from a positive number to a negative number and then continues to increase within the negative range.
23

---

Differential Coding for Training-Free ANN-to-SNN Conversion
Step 4: Analyze the Trend of the First Derivative (0)
Given:
((() -1 -1+ +ef erf
-1 V2a
<0, (+(00) = 0
(90)
Since (0) first decreases from a positive number to a negative number and then increases within the negative range, (0) will first increase from a negative number to a positive number and then decrease back zero.
Step 5: Analyze the Trend of the Function f(0)
Given:
S(0) = (1 (1-) Σ
-1
22
-H
erf
-
E
20
1
-rf
+
>0 0 (91)
22
V20
1=1
i=1
11
2i
(21-1)00
2n
11
)
f(+00) p(1 1)
0
1
erf
-0+0
(92)
T 2
n
n
220
i=1
1=1
Since (0) first increases from a negative number to a positive number and then decreases back to zero, f(0) will first decrease from a positive number to a minimum value and then increase towards zero
Therefore, (0) has unique root, which implies that the local optimal threshold 0 the global optimal threshold.
J. The Employed Neuron Model
We use a differential version of the Multi-Threshold (MT) neuron, as introduced in (Huang et al., 2024). The differential MT neuron is characterized by several parameters, including the base threshold 0, and a total of 2n thresholds, with n positive and n negative thresholds. The threshold values of the differential MT neuron are indexed by i, where X represents the i-th threshold value in the layer I: 2/ 8¹, 2/1
g¹
2 20
2n-1'
(93)
0
No+1 0¹, N1 No42
N/2
2
2n-1
Let variables ['(t), st]. ('(t), m [t). v v'(t], and (t, represent the input current, output spike of the i - th threshold, encoded output value, the membrane potential before and after spikes in the 1-th layer at time-step 1, and another membrane potential to record encoded input rate information, respectively. The dynamics of the MT neurons are described by the following equations:
r't m + ' -'-((t)
(94)
m [t m x² -1)
a: (14
(95)
m [t] = r'(t - 1] + I (t).
(96)
<|[1] MTH,, (m' [t],
(97)
x [1] = Σ s </1.
(98)
o m [c]] - x² x'({],
(99)
0,if Xan Am
MT (t),
1.elif i = arg minp x
Ap
(100)
0.else
When 1, this model reduces a differential IF neuron with negative threshold.
24

---

Differential Coding for Training-Free ANN-to-SNN Conversion
K. Result of Different Models on ImageNet Dataset
Table 3 and present the evaluation results for various CNN-based and Transformer- models. The variable 2n denotes the number of positive and negative thresholds in the multi-threshold neurons, where the negative thresholds are the opposite number of corresponding positive thresholds. The energy ratio is the energy consumption of SNNs divided by that of ANNs. For the ResNet18, ResNet34, and VGG16 models, the threshold scale is set to 1. For the ViT and EVA02 models, the threshold scale is 4.
Table 3. Accuracy and Energy Efficiency of DCGS(Ours) on CNN-based models for ImageNet Dataset
Accuracy
Time-step T
Architectun//Praaeetr((M)))
Origina(ANNN
Energy
2
4
8
16
32
64
Acc
0.10
0.11
1.57
51.89
69.89
71.08
Energy ratio
0.05
0.09
0.18
0.31
0.49
0.76
ResNet18 / 11.7M
71.49
Acc
61,45
70.07
71.31
71.47
71.49
Energy ratio
0.14
0.22
0.33
0,46
0.66
Acc
65.30
70.96
71.40
71.51
8
Energy ratio
0.17
0.32
0.48
0.63
Acc
0.10
0.14
0.46
8.76
58.86
74.11
Energy ratio
0.04
0.09
0.19
0.34
0.61
0.97
ResNet34 / 21.8M
76.42
Acc
59.71
73.35
76.04
76.35
76.38
Energy ratio
0.14
0.24
0.37
0.53
0.76
Acc
65.23
74.68
76.17
76.37
Energy ratio
0.18
0.34
0.55
0.75
Acc
0.08
0.16
1.03
55.48
72.04
73.13
Energy ratio
0.03
0.06
0.13
0.19
0.29
0.40
/GG16/ 138M
73.25
Acc
70.69
72.72
73.17
73.26
73.24
Energy ratio
0.10
0.15
0.22
0.29
0.38
Acc
72.26
73.16
73.22
73.22
Energy ratio
0.15
0.28
0.37
0.45
L. Effectiveness of Differential Coding
Table 5 presents comparative experiment between differential coding and rate coding. In most cases, differential coding outperforms rate coding in both accuracy and energy ratio, particularly increases.
M. Effectiveness of Threshold Iteration Method
Table 6 presents a comparative experiment between the threshold iteration method and the 99.9% large activation method. The threshold iteration method outperforms the 99.9% large activation method across different threshold numbers n and threshold scales.
N. Evaluation results of object detection task on the COCO dataset
We evaluated the performance our approach for object detection task on the COCO dataset using three different models provided by torchvision in various parameter settings, along with ablation studies, as shown in Table and 8. The result shows that both differential coding and Threshold Iteration method improves the network's performance.
25

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
O. Evaluation results of Semantic segmentation task on the PascalVOC dataset
Additionally, we evaluated our method for semantic segmentation task on the PascalVOC dataset using two different models provided by torchvision in various parameter settings, also conducting ablation experiments, as presented in Table 9 and 10. The result shows that both differential coding and Threshold Iteration method improves the network's performance.
P. Algorithm of MT Neuron on GPU
To enable fast execution on the GPU, we also design an efficient algorithm for each time step, as illustrated in Algorithm 3. This algorithm leverages the torch. f float 32 data type and takes advantage of the IEEE 754 sanlle-reccison floating- point format, where the exponential part has a bias value of 127, as an example.
Algorithm 3 Algorithm of MT Neuron GPU
Input: Total input and membrane potential m of MT neuron, number of thresholds parameter n.
Output: Output sum Ven [i] S(t] which is denote as spike_sum
3: Step I: Add input to membrane potential
4:
5: Step 2: Set mantissa to zero
6: int_tensor (m* 4/3) view (torch. int32)
7: mantisss mask = (1 << 23) -
8: int.tensor int.tensor
"mant cantissaaman mask
9: Step 3: Extract the exponential part of spike
10: exponent_mase
0xFF << 23
= (int.tensor 12: Step 4: Find the appropriate threshold to output
11: spike.exponee
ayponent.maski >> 23
13: spike.exponent = torch. where (spike.exponent
127 <= -n, torch. tensor (0,) 127 > 0, torch. (127,) spike.exponent))
torch. where (spike_exponert
14: Step 5: Construct a new exponential part to construet the spike output sum
int tensor (int tensor "exponent mask) (spike exponent 23)
spike_sum int.tensor. view (torch. float32)
17: Step 6: Reset membrane potential
18:
spike_sum
This algorithm leverages the IEEE 754 float32 format to efficiently compute spike outputs in the neuron model. At each time step, input is added to the membrane potential m, which is then scaled and cast to 'int32' to access the exponent field directly.
By masking out the mantissa and extracting the exponent, the algorithm quickly determines the magnitude of m. Using the bias value 127 and threshold parameter n, it determines which threshold apply for generating a spike. The exponent adjusted accordingly and used to reconstruct 'spike sum' as float, which is then subtracted from to complete the reset.
This approach avoids conditional branches and enables efficient bitwise operations and vectorization on the GPU, making it suitable for high-speed MT neuron simulations.
26

---

Differential Coding for Training-Free ANN-to-SNN Conversion
Table 4. Accuracy and Energy Efficiency of DCGS(OurS) on Transformer-based models for ImageNet Dataset
Accuracy
Time-step T
Architectur/PPraaeett((())
(6)))))(ANNN(6))
Energy
8
16
32
64
Acc
0.1
0.1
0.1
0.11
0.19
64.92
Energy ratio
0.00
0.001
0.008
0.06
0.28
1.34
Acc
0.1
0.16
62.76
79.25
80.95
4
VIT-Small / M
81.38
Energy ratio
0.03
0.12
0.45
1.06
2.06
Acc
44.59
78.15
81.02
81.44
Energy ratio
0.20
0.44
0.83
1.44
Acc
77.84
81.11
81.43
81.38
Energy ratio
0.32
0.62
1.05
1.71
Acc
0.10
0.12
29.36
80.78
83.70
Energy ratio
0.01
0.05
0.28
0.54
0.74
1.44
ViT-Base 86.6M
84.54
Acc
0.10
46.03
82.72
84.69
6
Energy ratio
0.05
0.20
0.52
1.04
Acc
80.34
83.98
84.23
84.27
Energy ratio
0.28
0.54
0.92
1.46
Acc
0.10
0.10
0.18
80.74
85.00
Energy ratio
0.00
0.01
0.09
0.45
0.92
Vir-Large 304.3M
85.84
Acc
0.12
78.99
84.76
85.59
Energy ratio
0.06
0.23
0.51
0.94
Acc
83.73
85,45
85.68
85.74
Energy ratio
0.24
0.46
0.80
1.33
Acc
0.09
0.15
0.88
65.75
78.46
Energy ratio
0.01
0.05
0.24
0.94
2.30
EVA02-Tiny 5.8M
80.63
Acc
0.32
52.86
77.72
80.04
6
Energy ratio
0.11
0.34
0.86
1.81
Acc
66.32
79.56
80.38
80.578
Energy ratio
0.29
0.66
1.26
2.28
Acc
0.09
0.10
0.14
34.86
82.66
Energy ratio
0.01
0.04
0.19
0.67
1.98
EVA02-Small 22.1M
85.73
Acc
0.14
30.83
82.20
85.48
Energy ratio
0.09
0.29
0.80
1.71
Acc
71.37
84.70
85.64
85.72
Energy ratio
0.28
0.63
1.21
2.17
Acc
3.25
81.00
87.86
Energy ratio
0.11
0.36
0.82
EVA02-Base 87.
88.69
Acc
84.62
88.16
88.46
Energy ratio
0.30
0.64
1.17
Acc
12.41
87.02
89.57
6
Energy ratio
0.13
0.39
0.84
EVA02-Large / 305.1M
90.05
Acc
88.25
89.72
89.90
Energy ratio
0.31
0.64
1.15
27

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
Table 5. Effective of Differential Coding
Architecture @ (A)g
Coding Type@
Accuracy
Time-ster step T
Threshold scale
Energy
2
8
16
32
64
Acc
0.10
0.14
0.46
8.76
58.86
74.11
Energy ratio
0.04
0.09
0.19
0.34
0.61
0.97
Differential
Acc
59.71
73.35
76.04
76.35
76.38
4
Energy ratio
0.14
0.24
0.37
0.53
0.76
Acc
65.23
74.68
76.17
76.37
8 Energy ratio
0.18
0.34
0.55
0.75
Acc
0.10
0.13
0.31
2.78
46.29
73.07
Energy ratio
0.02
0.05
0.13
0.25
0.46
0.77
Differential @ x2
Acc
46.1
69.53
75.77
76.33
76.43
Energy ratio
0.11
0.20
0.32
0.48
0.71
ResNet34@76...
Acc
72.03
76.24
76.39
76.41
8
Energy ratio
0.17
0.32
0.50
0.66
Acc
0.11
0.29
11.03
52.78
68.05
71.04
Energy ratio
0.04
0.11
0.26
0.55
1.11
2.22
Rate @ x
Acc
58.46
68.46
71.20
71.77
71.99
Energy ratio
0.15
0.31
0.62
1.22
2.42
Acc
59.79
68.61
71.1
71.8
8
Energy ratio
0.19
0.38
0.74
1.45
Acc
0.10
0.10
1.50
41.08
69.78
74.95
Energy ratio
0.02
0.07
0.17
0.38
0.77
1.54
Rate @ x2
Acc
51.34
71.22
75.11
75.78
75.92
Energy ratio
0.13
0.26
0.53
1.05
2.09
Acc
65.26
74.24
75.72
75.96
8
Energy ratio
0.19
0.46
0.73
1.43
Acc
0.1
0.1
0.1
0.11
0.19
64.92
Energy ratio
0.00
0.001
0.008
0.06
0.28
1.34
Acc
0.1
0.16
62.76
79.25
80.95
Differential @ x4
Energy ratio
0.03
0.12
0.45
1.06
2.06
Acc
77.84
81.11
81.43
81.38
Energy ratio
0.32
0.62
1.05
1.71
VT-Small@81.33
Acc
0.1
0.1
0.1
0.1
0.12
54.26
Energy ratio
0.00
0.001
0.008
0.05
0.22
1.17
Rate @ x4
Acc
0.1
0.14
51.46
78.07
80.69
Energy ratio
0.03
0.12
0.44
1.13
2.47
Acc
75.64
80.29
81.18
81.36
Energy ratio
0.32
0.67
1.38
2.81
28

---

Differential Coding for Training-Free ANN-to-SNN Conversion
Table Effective of threshold iteration Method
Architecture @ (AN
Find Threshold Method@
Accuracy
Time-step T
x XThreshold Scale
Energy
4
8
16
32
64
Acc
0.10
0.14
0.46
8.76
58.86
74.11
Energy ratio
0.04
0.09
0.19
0.34
0.61
0.97
threshold iteration x
Acc
59.71
73.35
76.04
76.35
76.38
Energy ratio
0.14
0.24
0.37
0.53
0.76
Acc
65.23
74.68
76.17
76.37
Energy ratio
0.18
0.34
0.55
0.75
Ace
0.10
0.13
0.31
2.78
46.29
73.07
Energy ratio
0.02
0.05
0.13
0.25
0.46
0.77
threshold iteration @ x2
Acc
46.1
69.53
75.78
76.33
76.43
Energy ratio
0.11
0.20
0.32
0.48
0.71
Res7et34@76.42
Acc
72.03
76.24
76.39
76.41
Energy ratio
0.17
0.32
0.50
0.66
Acc
0.10
0.14
0.74
7.08
45.68
69.50
Energy ratio
0.04
0.09
0.19
0.35
0.64
1.08
99.9% large activation @ x
Acc
34.08
63.18
74.41
75.82
76.14
Energy ratio
0.13
0.24
0.39
0.60
0.91
Acc
49.60
71.53
75.46
76.02
Energy ratio
0.18
0.33
0.55
0.78
Acc
0.10
0.10
0.22
0.93
31.32
71.36
Energy ratio
0.02
0.05
0.13
0.25
0.50
0.89
99.9% large activation @ x2
Acc
15.74
50.97
73.88
76.12
76.32
Energy ratio
0.11
0.20
0.35
0.55
0.86
Acc
67.92
75.54
76.25
76.40
Energy ratio
0.18
0.32
0.50
0.71
Table 7. Accuracy and energy efficiency of DCGS (Ours) across different models for object detection task on the COCO dataset.
mAP
Time-step T
Architecture
ANN MAP9[[1U==005000005
Energy ratio
4
8
mAP
0.0
0.2
1,6
6.3
FCOS_ResNet50
39.2
2
Energy ratio
0.12
0.24
0.35
0.47
mAP%
21.0
33.9
36.7
38.2
FCOS ResNet50
39.2
4
Energy ratio
0.16
0.31
0.43
0.55
mAP%
30.5
38.5
39.2
39.2
ECOS_ResNet50
39,2
8
Energy ratio
0.22
0.42
0.61
0.75
mAP%
25.6
33.9
35.8
36.0
Retinanet ResNet50
36.4
8
Energy ratio
0.23
0.44
0.63
0.78
mAP%
19.7
32.6
37.9
39.7
Retinanet ResNet50 v2
41.5
8
Energy ratio
0.22
0.43
0.64
0.84
29

---

Differential Coding for Training-Free ANN-1o-SNN Conversion
Table 8. Ablation Study of DCGS (Ours) on FCOS ResNet50 model for object detection task on the COCO dataset.
mAP
step T
Coding Type
Threshold Searching method
Energy ratio
2
6
mAP%
30.5
38.5
39.2
39.2
Differential
Threshold Iteration
Energy ratio
0.22
0.42
0.61
0.75
mAP%
21.8
31.5
34.3
35.5
Rate
Threshold Iteration
Energy ratio
0.22
0.44
0.66
0.88
mAP%
25.8
36.2
38.4
39.0
Differential
99.9% Large Activation
8
Energy ratio
0.22
0.43
0.62
0.78
Table Accuracy and energy efficiency of DCGS (Ours) across different models for semantic segmentation task on the PascalVOC dataset.
mloU
Time step T
Architecture
ANN mloU%
Energy ratio
2
6
mloU%
4.0
10.1
19.8
36.0
FCN ResNer50
64.2
Energy ratio
0.03
0.10
0.15
0.22
mloU%
51.8
60.5
62.7
64.0
FCN.ResNet50
64.2
Energy ratio
0.10
0.20
0.27
0.35
mloU%
61.0
64.3
64.6
64.5
FCN_ RCN_ResNet50
64.2
Energy ratio
0.18
0.34
0.50
0.63
mloU%
66.6
69.1
69.3
69.3
3.esNe00
69.3
Energy ratio
0.08
0.32
0.46
0.58
Table 10, Ablation Study of DCGS (Ours) on FC55ResNet50 model for semantic segmentation task on the PascalVOC datasct.
mAP
Time-step T
Coding Type
Threshold Searching Method
Energy ratio
2
4
8
mloU%
61.0
64.3
64.6
64.5
Differential
Threshold Iteration
8
Energy ratio
0.18
0.34
0.50
0.63
mloU%
58.2
62.9
63.7
63,9
Rate
Threshold Iteration
8
Energy ratio
0.18
0.37
0.54
0.71
mIoU%
61.2
64.3
64.5
64.4
Differential
99.9% Large Activation
8
Energy ratio
0.18
0.35
0.51
0.64
30


Reviews:
[{"review_id": "WiMEm6qs8y", "reviewer": "Reviewer_Kt6R", "summary": "ANN-to-SNN conversion has been known to produce so-called ‘conversion’ errors. Recent studies proposed methods that can reduce conversion errors, and in this study, the authors propose to improve the earlier studies with a novel algorithm named ‘differential coding’. Specifically, they focus on preventing the decay of the early spikes’ influences on spiking neurons’ outputs and show that a new encoding variable in spiking neurons can prevent neurons forgetting early spikes. The analytical analyses on neurons’ behaviors with differential coding makes this study’s objective clear, and the empirical evaluations are compelling. Given the novelty and potential influence of this newly proposed idea, I think this study may be of great interest to our readers.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive and constructive comments. We are delighted that you find our idea novel, interesting and well explained. We would like to address your concerns and answer your questions in the following.\n\n## 1. It would be great to see evaluations on a few more datasets to strengthen this study’s message.\nThank you for your suggestion to evaluate our method on more tasks to strengthen the message. We added the test results of our method on object detection and semantic segmentation tasks. \n### 1.1 Evaluation results of object detection task on the COCO dataset\nWe evaluated the performance of our approach for object detection task on the COCO dataset using three different models provided by torchvision in various parameter settings, along with ablation studies, as shown in the table below. The result shows that both differential coding and Threshold Iteration method improves the network's performance.\n\n**Table R1: Accuracy and energy efficiency of DCGS(Ours) across different models for object detection task on the COCO dataset.**\n| Architecture  | ANN mAP%\\[IoU=0.50:0.95\\]  \\/energy ratio | n | T=2   | T=4   | T=6  | T=8 |\n| -------- | -------------- | ----- | ----- | ----- | ----- | ----- | \n| FCOS_ResNet50| mAP%: 39.2| 2 | 0.0 | 0.2 | 1.6 | 6.3 |\n|              | energy ratio| 2 | 0.12 | 0.24 | 0.35 | 0.47 | \n| FCOS_ResNet50 | mAP%: 39.2 | 4 | 21.0 | 33.9 | 36.7 | 38.2 |\n|              | energy ratio    | 4 | 0.16 | 0.31 | 0.43 | 0.55 | \n| FCOS_ResNet50 | mAP%: 39.2 | 8 | 30.5 | 38.5 | 39.2 | 39.2 |\n|              | energy ratio    | 8 | 0.22 | 0.42 | 0.61 | 0.75 | \n| Retinanet_ResNet50| mAP%: 36.4| 8 | 25.6 | 33.9 | 35.8 | 36.0 |\n|              | energy ratio| 8 | 0.23 | 0.44 | 0.63 | 0.78 | \n| Retinanet \\_ResNet50_v2| mAP%: 41.5| 8 | 19.7 | 32.6 | 37.9 | 39.7 |\n|              | energy ratio| 8 | 0.22 | 0.43 | 0.64 | 0.84 | \n\n**Table R2: Ablation Study of DCGS(Ours) on FCOS_ResNet50 model for object detection task on the COCO dataset.**\n| Coding Type | Threshold Searching method | mAP% \\/energy ratio | n | T=2   | T=4   | T=6  | T=8 |\n| -------- | -------------- | ----- |----- | ----- | ----- | ----- | ----- | \n| Differential| Threshold Iteration |mAP%| 8 | 30.5 | 38.5 | 39.2 | 39.2 |\n|              | |energy ratio| 8 | 0.22 | 0.42 | 0.61 | 0.75 | \n| Rate| Threshold Iteration |mAP%| 8 | 21.8 | 31.5 | 34.3 | 35.5 |\n|              | |energy ratio| 8 | 0.22 | 0.44 | 0.66 | 0.88 | \n| Differential| 99.9% Large Activation |mAP%| 8 | 25.8 | 36.2 | 38.4 |39.0 |\n|              | |energy ratio| 8 | 0.22 | 0.43 | 0.62 | 0.78 | \n\n### 1.2 Evaluation results of Semantic segmentation task on the PascalVOC dataset \nAdditionally, we evaluated our method for semantic segmentation task on the PascalVOC dataset using two different models provided by torchvision in various parameter settings, also conducting ablation experiments, as presented in the table below. The result shows that both differential coding and Threshold Iteration method improves the network's performance.\n\n**Table R3: Accuracy and energy efficiency of DCGS(Ours) across different models for semantic segmentation task on the PascalVOC dataset.**\n| Architecture  | ANN mIoU% \\/energy ratio  | n    | T=2   | T=4   | T=6  | T=8 |\n| -------- | -------------- | ----- | ----- | ----- | ----- | ----- | \n| FCN_ResNet50| mIoU%: 64.2 | 2 | 4.0 | 10.1 | 19.8 | 36.0 |\n|    | energy ratio| 2 | 0.03 | 0.10 | 0.15 | 0.22 |\n| FCN_ResNet50 | mIoU%: 64.2 | 4 | 51.8 | 60.5 | 62.7 | 64.0 |\n|    | energy ratio    | 4 | 0.10 | 0.20 | 0.27 | 0.35 |\n| FCN_ResNet50 | mIoU%: 64.2 | 8 | 61.0 | 64.3 | 64.6 | 64.5 |\n|    | energy ratio   | 8 | 0.18 | 0.34 | 0.50 | 0.63 |\n| Deeplabv3_ResNet50| mIoU%: 69.3 | 8  | 66.6 | 69.1 | 69.3 | 69.3 |\n|    | energy ratio| 8 | 0.08 | 0.32 | 0.46 | 0.58 |\n\n\n**Table R4: Ablation Study of DCGS(Ours) on FCN_ResNet50 model for semantic segmentation task on the PascalVOC dataset.**\n| Coding Type | Threshold Searching method | mIoU% \\/energy ratio | n | T=2   | T=4   | T=6  | T=8 |\n| -------- | -------------- |----- | ----- | ----- | ----- | ----- | ----- | \n| Differential| Threshold Iteration |mIoU%| 8 | 61.0 | 64.3 | 64.6 | 64.5 |\n|    | |energy ratio| 8 | 0.18 | 0.34 | 0.50 | 0.63 |\n| Rate| Threshold Iteration |mIoU%| 8 | 58.2 | 62.9 | 63.7 | 63.9 |\n|    | |energy ratio| 8 | 0.18 | 0.37 | 0.54 | 0.71 |\n| Differential| 99.9% Large Activation |mIoU%| 8 | 61.2 | 64.3 | 64.5 | 64.4 |\n|    | |energy ratio| 8 | 0.18 | 0.35 | 0.51 | 0.64 |\n\n## 2. In lines 426-427, the authors write \"As shown in Table3, detailed results can be found in Appendix\". I think the authors may want to rephrase it for better readability.\nThank you for your suggestion. We will revise this sentence to \"The partial results are presented in Table 3, with a more detailed table provided in Appendix L.\". What's more, in the final version, we will reshape Table 3 and Table 5 into line charts to more intuitively compare the different methods.", "claims_and_evidence": "The paper proposes novel and interesting algorithms, which are well explained. Further, the empirical evaluations clearly support the utility of   the \"differential coding\" proposed by the authors.", "methods_and_evaluation_criteria": "The authors tested differentiable coding on only ImageNet. Since ImageNet is a golden standard for image classification, this may not be a critical issue, but it would be great to see evaluations on a few more datasets to strengthen this study’s message.", "theoretical_claims": "I did not find any issues with their description.", "experimental_designs_or_analyses": "Experimental designs and analyses are all sound.", "relation_to_broader_scientific_literature": "They described earlier studies sufficiently well.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "In lines 426-427, the authors write \"As shown in Table3, detailed results can be found in Appendix\". I think the authors may want to rephrase it for better readability.", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DPRrZ5ewf4", "reviewer": "Reviewer_Yuhw", "summary": "This paper introduces a novel differential coding scheme for training-free ANN-to-SNN conversion. The authors propose using time-weighted spikes as incremental updates rather than direct rate representations, significantly reducing energy consumption and spike counts. They detail an algorithmic framework integrating multi-threshold spiking neurons, differential coding for various layers (convolutions, fully connected, Transformers), and a threshold iteration method that optimally sets neuron thresholds under a normal distribution assumption.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your positive and thoughtful comments. We are encouraged that you find our method novel and well-explained, and our empirical results thorough. We would like to address your concerns and answer your questions in the following.\n\n### 1. Hardware implementation of MT Neuron and discussion about memory overhead and the relationship between differential coding and \"burst-like\" spiking behavior.\n\nThank you for your suggestion. We will first show how to implement a hardware friendly MT neuron, and then discuss the relationship between differential coding and \"burst-like\" spiking behavior.\n\n**Hardware implementation of MT Neuron**: \n\nCompared with previous ANN2SNN methods, the MT neuron is required to transmit an extra index $i$ for the threshold. When implementing the MT neuron on GPUs, two implementations can be considered:\n1. Sent $V_{th}[i] \\cdot S[t]$ to the next layer\n2. Add an external threshold dimension with $2n$ elements to $S[t]$, set $S[t][i]=1$ and $S[t][j]=0$ for all $j \\neq i$. At the same time, an external threshold dimension is added to the weight of the next layer, whose elements are the multi-level thresholds.\n\nFor simplicity, we use implementation 1 on GPUs, which is not pure binary but equivalent to implementation 2 with binary outputs. The MT neuron is also compatible with asynchronous computing neuromorphic chips because its outputs are still sparse events. Take the speck chip [1] as an example. The LIF neuron in the convolutional layer in speck chip outputs $(c,x,y)$ to the next layer (refer to Fig S4). When using the MT neuron, the only modification is adding a threshold index, i.e., $(c,x,y, i)$. The computations of the next layer should also be changed with a bit-shift operation on weights (because the threshold is the power of 2 and the multiplication is avoided). After the above modifications, the computation is still asynchronous and event-driven.\n\nThe implemention to avoid argmin in Equation 6 in hardware can be desctibed in the following two steps.\n\n**Step1:** Set all the base threshold $\\theta^l=1$ and get SNN weights by using the weight normalization strategy[2]. So, all thresholds in MT neuron are:\n$$\\lambda^l_i=\\begin{cases}\n\\frac{1}{2^{i-1}},&1<i\\leq n,\\\\\\\\\n\\frac{-1}{2^{i-n-1}},&n<i\\leq 2n.\n\\end{cases}$$\n**Step2:** \nWe define $\\frac{4}{3}m^l[t]=(-1)^{S}2^{E}(1+M)$ with $1$ sign bit ($S$), $8$ exponent bits ($E$), and $23$ mantissa bits ($M$).\nSince the median of $\\frac{1}{2^{k-1}}$ and $\\frac{1}{2^k}$ is $\\frac{3}{4}\\frac{1}{2^{k-1}}$, we can easily select the correct threshold index $i$ using $E$ and $S$ of $\\frac{4}{3}m^l[t]$, without performing $2n$ subtractions to calculate the argmin in Equation 6 :\n$$\\text{MTH}_{\\theta,n}(m^{l}[t],i)=\\begin{cases}1,&\\text{if }\\begin{cases}\ni<n,\\text{ S}=0\\text{ and }i=1-\\text{E},\\\\\\\\\ni\\geq n,\\text{ S}=1\\text{ and }i-n =1-\\text{E},\\end{cases}\\\\\\\\\n0,&\\text{otherwise}.\\end{cases}$$\nThe detailed implementation will be included in the final version.\n\nFor differential neurons, the memory overhead compared to initial neurons, such as IF or MT neurons, only includes an additional membrane potential. This extra potential is used to adjust the input current as described in Theorem 4.4.\n\n**The relationship between differential coding and \"burst-like\" spiking behavior**:\n\nIn this paper, due to the MT neurons, which select an appropriate threshold index to fire spikes, there is no short-time \"explosive\" accumulation problem. However, this does not prevent us from discussing the effects of using differential coding and burst coding on other neurons that have at least one negative threshold.\n\nUsing differential coding can significantly reduce the short-time \"explosive\" accumulation problem. Since the goal of ANN-to-SNN conversion is to approximate the activation values of an ANN, for neurons that initially suffered from this problem, the differential information diminishes over time. This gradual reduction ultimately eliminates the accumulation issue.\n\n[1] Spike-based Dynamic Computing with Asynchronous Sensing-Computing Neuromorphic Chip \n\n[2] Conversion of Continuous-Valued Deep Networks to Efficient Event-Driven Networks for Image Classification\n\n\n### 2. Disscussion on comparision with grid search and calibration-based approach.\n\nThank you for your suggestion. Due to time constraints, we plan to refine the code for the grid search and calibration-based methods in future work to conduct a more detailed comparison of their accuracy. However, **from a speed perspective**, our method calculates the theoretically optimal thresholds for all network modules—whether at the layer-wise, channel-wise, or neuron-wise level—within seconds after computing the mean and variance. We believe this represents a significant improvement in speed compared to grid search and calibration-based methods.\n\n\n### 3. Incomplete related works.\n\nThank you for your suggestion. We will cite {Adaptive Calibration [AAAI 2025]} and discuss this article in the final version", "claims_and_evidence": "The authors claim that differential coding reduces the spike rate and preserves high accuracy in converted SNNs. They support this claim with extensive experiments demonstrating both reduced energy consumption and competitive accuracy compared to baseline methods.", "methods_and_evaluation_criteria": "The proposed methods—differential coding and threshold optimization—are well-suited to ANN-to-SNN conversion tasks. The chosen evaluation criteria (accuracy and energy-related metrics) are appropriate and align with typical benchmarks for spiking networks.", "theoretical_claims": "The authors provide proofs and derivations in the supplemental material. The mathematical steps appear sound.", "experimental_designs_or_analyses": "The experimental designs—using multiple CNN and Transformer benchmarks—are comprehensive. The analysis is carefully presented, with comparisons to standard baselines and ablation studies showing how each component (e.g., differential coding vs. threshold tuning) contributes to overall performance.", "relation_to_broader_scientific_literature": "The paper builds on established ANN-to-SNN conversion methods but improves them through more efficient coding strategies.", "essential_references_not_discussed": "The paper covers the main works on ANN-to-SNN conversion and multi-threshold neurons. However, Adaptive Calibration [AAAI 2025] is also a training-free and multi-threshold neuron framework. I think it should be included.\n\nAdaptive Calibration: A Unified Conversion Framework of Spiking Neural Network [AAAI 2025]", "other_strengths_and_weaknesses": "Strengths:\n1. The proposed differential coding framework is novel and well-explained.\n2. The threshold iteration method is carefully justified with theoretical derivations.\n3. Empirical results are thorough, showing gains in both accuracy and energy efficiency.\n\nWeaknesses:\n1. While the authors do address threshold tuning, more empirical comparisons (e.g., grid-based calibration [1]) would help quantify speed and accuracy trade-offs.\n2. Discussions on hardware implementation are relatively brief—expanding on how differential coding would map to neuromorphic hardware (especially regarding memory overhead and potential short-time “explosive” accumulations) would clarify practical feasibility.\n\n[1] A free lunch from ann: Towards efficient, accurate spiking neural networks calibration.", "other_comments_or_suggestions": "1. It would be beneficial to include a small ablation experiment on threshold selection methods (e.g., comparing the threshold iteration method to a grid search or calibration-based approach) to show how quickly and accurately each method converges.\n\n2. A dedicated subsection on hardware aspects—whether differential coding might exacerbate or mitigate “burst-like” spiking under certain time scales—would strengthen the paper’s real-world applicability.", "questions_for_authors": "See Suggestions and Weaknesses", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4Nlptv7AYc", "reviewer": "Reviewer_QRfu", "summary": "In this work, the authors proposed differentiable neural coding. Based on the proposed coding, the authors provided differential graded units, differential spiking neurons, and differential coding for linear layer. According to the authors’ experiments, they could achieve state-of-the-art accuracy on image classification tasks.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful and very detailed feedback. We are delighted that you find our paper theoretically supported well and the results state-of-the-art. We would like to address your concerns and answer your questions in the following:\n\n### 1. Answer to  \"MT neurons are more complex to implement than LIF-series neurons. In particular, 2n subtract ($x-\\lambda_p$) operations are required to obtain the argmin of Equation 6.\" and \"Analysis of overheads such as proposed methods and MT neurons is required.\"\n\nThank you for raising these concern. Equation 6 in the article is presented for ease of understanding. In hardware implementation, the argmin module is not used. We have developed a hardware-friendly version of the MT neuron model, which can efficiently map the appropriate threshold using the potential's sign bit and exponent bits at an extremely low cost. **The detailed implementation of MT neuron can be seen in our response to Reviewer Yuhw.** We look forward to your review.\n\n### 2. Answer to \"It seems that the authors did not use binary spike activation.\" and explanation of  \"differential coding only updates the encoded activation value when an output spike occurs, rather than decay at each time-step in rate coding.\"\n\nWe use binary spike activation between neurons, as shown by Equation 6 and the red line in Figure 3a.\nEquations 7 and 12 respectively represent the encoding of the information sequence $x^l[1:t]$ in layer $l$ under rate coding and differential coding. Here, $x$ can be the weighted spike according to Equation 4. In rate coding, even though no spike is fired at a given time step and no explicit additional computation is performed, the meaning $r[t]$ encoded by the sequence $x^l[1:t]$ changes due to the increase in $t$, as Equation 7. Conversely, differential coding ensures that when no spike occurs at time-step $t$, the sequence $x^l[1:t]$ retains the same meaning as $x^l[1:t-1]$.\n\n### 3. From the experimental results in Table 1, it is judged that the CNN model using ReLU does not need minus vth. Why did MT neurons also use these models?\n\nDoes the term \" minus vth\" in your question refer to negative thresholds? Negative thresholds help minimize excessive spikes, which reduce unevenness errors[1] in conversion error. When converting a CNN model with ReLU, Differential Graded Units are not utilized. Instead, the ReLU activation function is replaced with a specific MT Neuron, which uses an additional mask to dynamically disable certain negative thresholds, ensuring the total output remains positive. Detailed implementation will be included in the appendix.\n\n[1] Optimal ANN-SNN Conversion for High-accuracy and Ultra-low-latency Spiking Neural Networks.\n\n### 4. Explain Equation 15 - How can the membrane potential ($m^l[t]$) and the firing rate ($r^{l-1}[t]$) of the previous layer be the same?\n\nTo unify the representation, we consider the linear layer as an independent layer, rather than just weights between layers. When layer $l-1$ is a linear layer, the actual previous layer should be $l-2$. In this case, $x^{l-1}[t]= W^lx^{l-2}=\\sum_iW^l\\lambda^ls^{l-2}$. Here, $r^{l-1}[t]$ represents the rate obtained by converting differencial coding back into a rate coding after linear layer.\n\nIn Equation 15, we define $m^l[t]=r^{l-1}[t]$ and further derive $m^l[t]=m^l[t-1]+\\frac{x^{l-1}[t]}{t}$.\n\n### 5.Explain Equation 16 - It is not reasonable to use non-linear activation (F) to approximate spiking neurons. It is reasonable to simulate only the behavior of spiking neurons that can be supported by neuromorphic hardware without non-linear activation.\n\nWe consider the non-linear activation as a part of the neuron's internal dynamics, separate from the communication between neuron layers. This design ensures that spike communication between neuronal layers remains uninterrupted, making our method feasible for hardware implementation. Furthermore, our Equation 16 specifies the expected output, which can also be approximated using multiple IF neurons to achieve hardware implementation[2].\n\n[2] Spatio-temporal approximation: A training-free snn conversion for transformers.\n\n### 6. Ablation studies on proposed methods are required. And the experimental results of object detection, segmentation, etc., if added, will be able to highlight the utility of the proposed method.\nThe ablation studies comparing differential coding and rate coding, as well as threshold iteration and the 99% large activation method, are detailed in Section 5 and Appendices K and M. If there’s anything further you’d like us to add, please let us know.\n\n**In response to Reviewer Kt6R, we have added new experiments and ablation studies on object detection and semantic segmentation tasks.** We look forward to your review.\n\n### 7. Refine Figure 1 and Table 1.\n\nThank you for your suggestion. We will revise Figure 1 to make it more biologically explainable and update Table 1 to ensure it is easier for comparisons.", "claims_and_evidence": "The proposed methods are theoretically supported well. Experimental evidence should be added to improve the paper.", "methods_and_evaluation_criteria": "The most important limitation of the SNN model is its applicability to neuromorphic hardware. SNNs must operate on event-based neuromorphic hardware to ensure low-power operation. From this perspective, the method proposed in this study has major concerns. There are great doubts about whether the proposed differentiable neural coding can be applied to neuromorphic hardware. There are also concerns about the feasibility of implementing the proposed differential graded units, differential spiking neurons, and differential linear layers derived from neural coding on neuromorphic hardware. The proposed method is expected to be difficult to implement on neuromorphic hardware.\n\nMT neurons are more complex to implement than LIF-series neurons. In particular, 2n subtract (x-\\lambda_p) operations are required to obtain the argmin of Equation 6. This computational overhead not only offsets the advantages of SNNs operating on neuromorphic hardware, but may also make implementation impossible. How can MT neurons be implemented on neuromorphic hardware? - How can it be applied to LIF?", "theoretical_claims": "The biggest difference between the existing rate coding (Equation 7) and the proposed coding (Equation 14) is described in the manuscript as “differential coding only updates the encoded activation value when an output spike occurs, rather than decay at each time-step in rate coding.” (line 205~). However, this is based on an incorrect fact. Equation 7 is only an equation to explain the ANN-to-SNN conversion of rate coding, and the actual output of the SNN is a binary spike train. Therefore, as the authors mentioned, the firing rate is not calculated at every time point in the spiking neuron, but binary spike output is generated only when a spike is fired. In this regard, there is no difference between the proposed coding and rate coding. Based on the authors’ claim, it seems that the authors did not use binary spike activation. If so, this greatly worsens the advantages of SNN for event-based computing. If binary spike activation is not used, how can it be utilized in neuromorphic hardware? If it is difficult to utilize in neuromorphic hardware, it seems reasonable to consider the proposed model as a DNN with a new activation function.\n\n- Equation 15 - How can the membrane potential (m^l[t]) and the firing rate (r^(l-1)[t]) of the previous layer be the same?\n\n- Equation 16 - It is not reasonable to use non-linear activation (F) to approximate spiking neurons. It is reasonable to simulate only the behavior of spiking neurons that can be supported by neuromorphic hardware without non-linear activation.", "experimental_designs_or_analyses": "Ablation studies on proposed methods are required.\n\nFrom the experimental results in Table 1, it is judged that the CNN model using ReLU does not need minus vth. Why did MT neurons also use these models?\n\nAnalysis of overheads such as proposed methods and MT neurons is required. Also, these overheads should be considered in comparing energy consumption.\n\nIn addition to image recognition, the experimental results of object detection, segmentation, etc., if added, will be able to highlight the utility of the proposed method.", "relation_to_broader_scientific_literature": "It will help advance neuromorphic computing.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Please refer to the above comments.", "other_comments_or_suggestions": "- It would be better to add synapse part to Figure 1.\n- For convenient comparisons, It would be better to present the accuracy of ANN for each experimental result in Table 1. (Even if it is also in the supplementary)", "questions_for_authors": "Please refer to the above comments.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose a so-called differential coding algorithm for converting ANN networks to spiking neural networks (SNN), by transmitting changes to the rate rather than the actual spike count. 
The authors also present a threshold iteration method for optimizing the spiking thresholds when converting ReLUs to spiking neurons.
The authors do various experiments on CNNs and Transformer architectures and argue that the algorithms
can improve or maintain accuracy while reducing energy consumption.
Overall the paper received 3 reviews, 2 of which were positive and 1 was slightly negative (weak reject).
The positive comments were related to the novelty of the approach, the theoretical justification for the algorithm and the thoroughness of the tests.
One of the reviewers questioned the applicability of the algorithm to current neuromorphic hardware architectures and questioned the efficiency of the proposed neuron model. There was no response to authors rebuttal related to this comment.
Overall I am leaning towards accepting the paper, but I suggest the authors take advantage of the reviewer comments to improve future iterations of the paper.
Decision: Accept (poster)