Dendritic Resonate-and-Fire Neuron for Effective and Efficient Long Sequence Modeling
Authors: Dehao Zhang
Malu Zhang
Shuai Wang
Jingya Wang
Wenjie Wei
Zeyu Ma
Guoqing Wang
Yang Yang
Haizhou Li
Venue: NeurIPS 2025 poster ()
Keywords: Resonate and Fire Neuron
Long Sequence Modeling
Spiking Neural Networks
Primary area: neuroscience_and_cognitive_science
TL;DR: Inspired by the dendritic structure of biological neurons, we propose a Dendritic Resonate-and-Fire (D-RF) model for effective and efficient long sequence modeling.

Abstract: The explosive growth in sequence length has intensified the demand for effective and efficient long sequence modeling. Benefiting from intrinsic oscillatory membrane dynamics, Resonate-and-Fire (RF) neurons can efficiently extract frequency components from input signals and encode them into spatiotemporal spike trains, making them well-suited for long sequence modeling. However, RF neurons exhibit limited effective memory capacity and a trade-off between energy efficiency and training speed on complex temporal tasks. Inspired by the dendritic structure of biological neurons, we propose a Dendritic Resonate-and-Fire (D-RF) model, which explicitly incorporates a multi-dendritic and soma architecture. Each dendritic branch encodes specific frequency bands by utilizing the intrinsic oscillatory dynamics of RF neurons, thereby collectively achieving comprehensive frequency representation. Furthermore, we introduce an adaptive threshold mechanism into the soma structure. This mechanism adjusts the firing threshold according to historical spiking activity, thereby reducing redundant spikes while maintaining training efficiency in long-sequence tasks. Extensive experiments demonstrate that our method maintains competitive accuracy while substantially ensuring sparse spikes without compromising computational efficiency during training. These results underscore its potential as an effective and efficient solution for long sequence modeling on edge platforms.

Dendritic Resonate-and-FFre Neuron for Effective and Efficient Long Sequence Modeling
Dehao Zhang', Malu Zhang 2 Shuai Wang', Jingya Wang', Wenjie Wei', Zeyu Ma', Guoqing Wang', Yang Yang', Haizhou Li
University of Electronic Science and Technology of China
Shenzhen Loop Area Institute, 'The Chinese University of Hong Kong (Shenzhen)
Abstract
The explosive growth in sequence length has intensified the demand for effecttvee and efficient long sequence modeling. Benefiting from intrinsic oscillatory mem- brane dynamics, Resonate-and-Fire (RF) neurons can efficiently extract frequency components from input signals and encode them into spatiotemporal spike trains, making them well-suited for long sequence modeling. However, RF neurons hibit limited effective memory capacity and a trade-off between energy efficiency and training speed on complex temporal tasks. Inspired by the dendritic structure of biological neurons, we propose a Dendritic Resonate-and-Fing (D-RF) model, which explicitly incorporates multi-dendritic and soma architecture. Fach den- dritic branch encodes specific frequency bands by utilizing the intrinsic oscillatory dynamics RF neurons, thereby collectively achieving comprehensive frequency representation. Furthermore, we introduce an adaptive threshold mechanism into the soma structure. This mechanism adjusts the firing threshold according to histor- ical spiking activity, thereby reducing redundant spikes while maintaining training efficiency in long-sequence tasks. Extensive experiments demonstrate that our method maintains competitive accuracy while substantially ensuring sparse spikes without compromising computational efficiency during training. These results underscore its potential as an effective and efficient solution for long sequence modeling on edge platforms.
Introduction
Long sequence modeling efficiently captures complex temporal patterns and dynamic characteristics, demonstrating exceptional application potential in edge computing scenarios such as speech recog- nition [35
(65) and electroencephalogran (EEG) monitoring 48) 67]. However, mainstream sequence modeling methods still primarily rely on Recurrent Neural Networks (RNNs) 44 [2].
64
Transformers 29 62 and state-space models (SSMs) 18 Although these approaches effectively compress contextual information into finite states, they still involve extensive floating- poating-pointt matrix multiplications, resulting high computational complexity, inference latency, and energy consump- tion [5]]. Therefore, designing long sequence models that simultaneously achieve high performance, energy efficiency, and fast inference remains an essential and ongoing research challenge.
Inspired by the structure and function of neural circuits in the brain, Spiking Neural Networks (SNNs) have emerged as biologically plausible and computationally efficient model (37
59
Unlike
Artificial Neural Networks (ANNs), SNNs possess event-driven computational capabilities 77 78
and the potential to process dynamic temporal information [72 70 80]. These properties allow information to be transmitted through binary spikes and enable the retention of historical context
CCoresponding author: maluzhang @westc.edu.crr
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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via membrane potentials. Several studies have applied SNNs for long sequence modeling based
on Leaky Integrate-and -nn-FFer (LIF) neurons. Zhang et al. [79 introduce two-compartment LIF neuron to better capture long-term temporal dependencies, while Fang et al. [14] remove the reset mechanism to maximize the utilization of information across all timesteps. However, due to their serial charge-fre--reett dynamics, these methods fail to capture complex timescale variations and
long-range dependencies in long sequences [66] 72 76), limiting their performance in complex tasks.
As an efficient alternative to LIF neurons, Resonate-and-Fire (RF) neurons 27 employ complex- valued state variables to more effectively retain historical information in long sequences modeling. Higuchi et al. [21] integrate refractory mechanism into the reset dynamics of RF neurons, main- taining frequency selectivity while significantly improving long sequence processing capacity and promoting spike pattern sparsity. Huang et al. [24] remove the reset mechanism and propose that a learnable time constant can capture the intrinsic reset behavior of RF neurons. This approach reduces the computational complexity from (C2) to (L L), thereby substantially improving compu- tational efficiency. Despite these advancements, current methods still face two major challenges. First, the limited bandwidth of RF neurons constrains their ability to extract diverse frequency- -band combinations from complex temporal signals, making them behave more like simplified resonators. Second, RF neurons encounter trade-off between energy efficiency and training speed: removing the reset mechanism enables efficient training but leads to excessive spike activity, whereas incorporating reset suppresses spiking activity at the cost of increased training overhead.
26] we propose novel Dendritic Resonate-and-fiee (D-RF) neuron for effective and efficient long sequence modeling. It consists of a multi-dendrite and soma structure. First, each dendritic branch captures specific frequency responses from input signals through the oscillation characteristies of RF neurons, thereby achieving comprehensive spectral decomposition across multiple timescales. Second, we incorporate an adaptive threshold mechanism into the soma structure that dynamically adjusts thresholds based on historical spiking patterns, achieving sparse spikes while maintaining training efficiency. Extensive experiments on long sequence tasks confirm the nigh-performance and energy-efficient of the proposed neuron model. The main contributions are summarized as follows:
Inspired by the dendritic structure of biological neurons 44
38
We conduct a detailed analysis of the limitations of existing RF neurons in long sequence modeling, highlighting their restricted memory capacity and the inherent trade-off between energy efficiency and training speed. First, the limited bandwidth response causes RF neurons behave like simplified resonators. Second, the presence absence of reset mechanism leads to a conflict between sparse spiking and training efficiency.
We propose the D-RF neuron, which comprises two components. First, dendritic branches exploit RF dynamics to achieve specialized frequency selectivity, collectively enabling full spectral coverage across multiple timescales. Second, an adaptive threshold mechanism in the soma dynamically adjusts thresholds based on history spiking activity, balancing computational cost and energy efficiency while preserving training effectiveness.
Extensive experiments demonstrate that our method achieves competitive performance across various long sequence tasks. Moreover, the method produces sparser spiking activity while maintaining training efficiency. These results highlight our model's dual advantages in effectiveness and computational efficiency for long sequence modeling.
2
Related Work
2.1 Advanced Spiking Neurons for Long Sequence Modeling
Due to the dynamic characteristics of spiking neurons, it is believed that they have the ability to handle
long sequence modeling. However, the LIF model [17 28 and its variants 13 72] exhibit limited memory capacity in temporal tasks, which is considered a critical factor for effective long sequence modeling. To overeome this limitation, several studies 7 [9] draw inspiration from more complex neural dynamics. RF neurons [27] have attracted considerable attention due to their intrinsic frequency band preference. Orchard et al. [4] leverage RF neurons to encode raw signals into sparse spike trains, thereby significantly reducing output bandwidth. Furthermore, Higuchi et al. [2]] introduce adaptive decay factor mechanisms [50 [6] and refractory mechanisms [54], improving the balance between energy efficiency and performance of RF neurons in long-range sequence modeling. In addition,

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RF-based models demonstrate competitive performance in sequence modeling tasks, including image
classification [22]. optical flow tracking [1]] and audio processing [53] 75]. Moreover, RF neurons
can be efficiently implemented on neuromorphic hardware like the Loihi 9 41
2.2 Training Strategies for Spiking Neural Networks
The mainstream training methods for deep SNNs can be categorized into ANN-to-Sie to-SNN conversion [2
47 63] and direct training [68] [69]. ANN-1o-SNN conversion methods use the similarity between spike firing rates and ANN activation functions, but need many timesteps to reach high accuracy. In contrast, direct training enables SNNs to achieve performance comparable to ANNs with the same architecture within limited number of timesteps. Specifically, direct training introduces surrogate gradient functions [T] 40 to enable backpropagation, thereby addressing the non-differentiabiiiy of spike firing functions. However, applying direct training to long sequence tasks presents greater challenges [5]], as such tasks often require thousands of timesteps. Consequently, some research [23 58 [66] tend to explore more efficient training strategies for SNNs. Yin et al. [73] further explore intrinsic challenges in SNNs training, especially the reset behavior of neurons. Therefore, Fang al. [14] change the dynamic process of spiking neurons into a learnable matrix, avoiding the reset mechanism. Similarly, Shen al. [51] introduce the SDN block simulate the reset process. These methods greatly reduce the training cost of SNNs while keeping their asynchronous inference ability.
3 Preliminary
3.1 Resonate-and-Fire Neuron
Inspired by the damped and sustained subthreshold oscillations observed in the membrane potentials of mammalian nervous systems 2 42 46), RF neurons are proposed (27]. Given input signal
I(t), the dynamics of the RF neuron timestep & can be described as follows:
dt (t) = (b +t)=(t)))))))))) + I(t).
(I)
=u E C represents the complex state of RF neurons, where represents current-like variable capturing voltage-gated and synaptic current dynamics, and the imaginary component v corresponds to voltage-like variable. w > 0 denotes the angular frequency of the neuron, indicating the number of radians it oscillates per second, while the damping factor the oscillation. It can be discretized using the Euler method [3]
regulates the exponential decay of
=|4 exp (ō(b + iw)) = 1] + 8Z[[]:
ô is the discrete timestep. When the real part of z[t] exceeds the threshold, the neuron fires spike; otherwise, it remains silent. Additionally, RF neuron exhibits preference for specific frequency bands. As shown in Fig. Ma). we present the oscillatory behavior under spike inputs with different frequencies. It observed that both the membrane potential and the phase state accumulate rapidly.
3.2 Direct Training in Spiking Neural Networks
Due to the BPTT 68 and Surrogate Gradient methods 40. training large-scale SNNs becomes feasible. Specifically, the gradient of the weight w' at timestep T can be represented as follows:
OL Ovd
C
s- st)
where ut (t) and (t) denote the membrane potential and spike emission of the I-th layer at time t, respectively. L is the loss function. It can be calculated as follows:
Oc ac os'( (t)
OC
Oud
(4)
Oi)
(t)
Ot Oul(1)
ac ac 0ul+1(1) ac
Ou (1+
(5)
()((()
Ou +1)+ (1 Du 11) (s
3

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The condiifferentiable term (t'(t) can be substituted with surrogate function. the (h++1 85 and Ou (n'(
(t)
are the temporal gradients that need to be calculated. As shown Eq. 4 and Eq. S the gradient at each timestep depends not only on the current states but also recursively on future states. This
recursive dependency across layers and timesteps results in a computational complexity of (
Method
4.1 Problem Analysis
Limited Performance in Complex Tasks: Benefiting from the design of the decay kernel and complex-valued states, RF neurons exhibit pronounced frequency selectivity. However, this property also limits the model's ability to discriminate diverse input patterns. As shown in Fig. [b), input signals with distinct frequency components may elicit similar spiking responses, as components misaligned with the neuron's 's intrinsic frequency tend to be suppressed. thereby impairing the network's capacity to capture and distinguish diverse temporal features. To further verify this limitation, we visualize the frequency response of a single RF neuron. The results show that the neuron responds primarily within narrow bandwidth and reaches its peak the intrinsie frequency, making difficult to capture complex frequency compositions. This observation highlights an inherent representational limitation of RF neurons in modeling complex temporal features.
(a) Vanilla Resonate Fire Model Response I;
(b) Limit Ability to Time Series
(c) Energy Efficiency vs. Training Speod
Response l2 G + la
Same
(I) Time-variant Hinders in Parallelization
Vnn
Vih
RF Neuron
Spike
H
Vin.
40 11
O(L
1
30
(2) Time-invariant Stuck Continuous Fire
Phuse Diagram
Phase Dhugeau
U2
Different Frequency Responses
* 00
Vin
(4-1
40
((Llng
Corr Kernal
50
100
14n
200
200
x1-11
Frequency
Charge
fire
Update
Backward
4. Frequency-Matchen Spike Trains to RF
1z: Frequency- Mingeaccy--ssmttcheed Spike Truins to RF
Figure 1: Problem Analysis: (a) Response of Different Spike Trains: Frequency-matched inputs
lead to rapid membrane potential accumulation, whereas mismatched inputs yield weaker responses. (b) Limited Ability to Time Series: A single RF neuron struggles to respond to different frequency- varying inputs due its narrow band selectivity. (c) Energy Efficiency vs. Training Speed: The time-variant method enables sparse spiking but has high training cost of O( The method allows faster training with complexity of OC L), but often leads continuous fire.
Challenges in Balancing Energy Efficiency and Training Speed: In long sequence tasks, RF neurons incur significant training costs and potentially excessive energy consumption. Existing research focuses on two main strategies. First, as shown in Fig. [c). Higuchi et al. [21] propose an adaptive decay factor that instantly raises the decay coefficient after each spike, suppressing membrane potential oscillations and promoting sparse spikes. However, this approach requires temporal unfolding during backpropagation, resulting ((C)) complexity. Second, Huang et al. (4 employ learnable decay factor to the reset mechanism and use Fourier transform reformulation to reformulate the temporal dynamics into a parallel convolutional process between an oscillatory kernel and the input signal, redueing computational complexity to O log C). Nevertheless, due to the decay factor remains time-invariant, the model preserves its pre-spike amplitude. As shown in Fig. 2, it results in sustained burst firing that undermines the low-power advantages of SNNs.
4.2 Dendritic Resonate-and-FF Fire Neuron
To better capture features across different frequency bands, we propose the D-RF neuron model. Unlike the vanilla RF model [27]. D-RF model comprises soma and multiple dendritic branches. Each dendritic branch extracts state responses corresponding to specific frequency preferences in the input signal Z[t]. When Z[t] contains frequency components aligned with branch's 's preference, the

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membrane potential of that branch accumulates rapidly. The soma integrates input currents from all dendritic branches and generates spike once its membrane potential exceeds predefined threshold. Specifically, the membrane potential dynamics of the i-th dendritic branch is defined as follows:
dz+(t) (-1/T; + iw)) 21) + 7II()), dt
where 7; and wi represent the decay factor and membrane potential oscillation coefficient associated with the i-th dendritic branch, respectively. I(t) denotes the presynaptic input at time t, and % represents the membrane capacitance of the i-th dendritic branch. This modeling framework allows different dendritic branches to selectively respond to specific frequency components. To enable efficient inference, the Zero-Order Hold (ZOH) method (IO] is employed for discretization. The membrane potential dynamics of all dendrites are given by:
kw
71 0
122
Z exp
8
Zlt [+T.]
(7)
1 iwn.
Here, 8 denotes the discrete timestep, and 2 22, ] represents the states of individual dendritic branches, with I' = [11, 72. 7m denoting their respective time constants. To further enhance the frequency characteristics across dendritic branches, each branch is assigned an individual importance weight. The dynamics of soma are defined as follows:
H = (Z)),,
S(4 = θ (H() - V4,
E [ denotes the importance weights assigned to each dendritic branch. O(.) represents the Heaviside function. When the presynaptic membrane potential of the soma [t] exceeds the threshold Vih. spike S[I] generated.
We further analyze the effectiveness of the dendrite design by examining its frequency band responses. For time-invariant RF neuron with decay kernel + iw, it can be modeled as a time-invariant convolutional process with kernel h(n), defined as h(n) = exp + iw))". Detailed proof is provided in the Appendix. A frequency response is described as follows:
2o
[[((ex(((0)]I = Σ h(m) exp( -i5mm = - exp + i(ōw 2]}
Therefore, a single RF neuron can be regarded as a first-order band-pass filter with resonance peak at Ω x w, and a narrow frequency band determined by the damping factor åb (as shown in Fig, (b)) In contrast, the frequency response our D-RF model is defined as follows:
[[(oor((ex((ii)]]l Σ C. (exp(((ex)())) Betf
B
(10)
Bett denotes the total frequency response of the D-RF neuron, while 6, [0, 1) quantifies the inde- pendent contribution of the i-th branch the overall frequency coverage. Consequently, compared to its mangl-brrach counterpart, the D-RF neuron provides a significantly broader frequency sensitivity.
4.3 Adaptive Threshold for Accelerated and Efficient Learning
To balance training speed and energy efficiency, we propose an adaptive thresholding strategy that dynamically adjusts the threshold based on the spiking activity from previous timesteps. Specifically, the threshold at timestep is defined as:
[t Σ a ORR(ZZ 1s 1...., 11} Vpre) ) Vpres
(11)
where VTR denotes the origin threshold (set to 1), and as E (0, represents the importance of preceding spikes. As shown in Fig, 2b) a% is the shared parameter during the adaptive threshold
5

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(n) Dendrite Sesonatt-und-Fing Neuron
(b) Adaptive Threshold
(c) Parallel Computation
1s[t]
Dendrite Branch
Dynamic of Branch;
Time Domain
Iogut
D. T
P
hid
[t
HIL
Soma
1
Integration
Different Brunc Kemel
Threshold at
1
Heaviside Function
[2
1D
Ougut Spiking Firing
Heaviside Function
1 1
Branch, Branch
Branchs
Branch
Casual Conv
Muldi-eendrites and Soma Structure
Venu 2 Sp
Thrcsbolds all timosleps
[12 [+]
Reser
Spike
H[Q]
9/ (t)
Soma
Stt]
211,2....
4. Spike
Silent
S Adaptive Threshold
Input
Output
Figure The Structure of D-RF model: (a) Dendrite Resonate-and-Fire Neuron: D-RF model consists of multiple dendritic branches, each encoding frequency-specite dynamics. The soma integrates membrane potentials from all branches through an adaptive threshold mechanism to enable sparse spikes. (b) Adaptive Threshold: The threshold at time is dynamically determined based on the historical spiking activity. (c) Parallel Computation: Different convolution kernels on each branch and causal convolution for the adaptive threshold allow parallel processing of the input over time.
computation. This process can be interpreted as one-dimensional causal convolution with kernel size where the kernel is defined as A = [at, On.. The spiking process can be reformulated as:
Sit] O C'R(2)
(Conv (d R(2) - Vpre) + Vpre [t]
(12)
Dendritic Input
Adaptive Threshold
Convid(.) present causal convolution along the temporal domain, enabling sparser spike activity while preserving the parallelizable nature of training. We demonstrate the effectiveness of this strategy by analyzing both the forward and backward propagation processes.
In the forward propagation, the computational complexity of the D-RF neuron is determined by multi-dendritic input and an adaptive threshold mechanism. Without the need for temporal unfolding, the D-RF neuron be formulated in parallelized manner. For the component of dendritic input, each dendritic branch functions independently and is decoupled in time. Therefore, the membrane potentials of all branches at time are defined as follows:
z' [1] Ei=o I' exp (k- .DD) T' [itt-k] k),
(13)
D characterizes the oscillatory resonators of individual dendritic branches and is defined as a diagonal matrix: D Diag (-1/n + w1, -1/72 w2,
-1/T,, + w,). Since D is fime-invariant, Eq. 13 can reformulated as convolution between the input signal I and the kernel K:
Z (K * [t] = F- (F(K)) F(I+} [t], K= 8D 8D2
8D"),
(14)
F(-) and F (-) denote the forward and inverse Fourier transform operations. Consequently, the charging process ensures a computational complexity of OC C) without introducing additional training overhead. For the adaptive threshold mechanism, its implementation via convolution opera- tions well-suited for GPU acceleration. Additionally, since ag 0, greater number of spikes in the previous timestep increases the threshold at time t, thereby promoting sparser spikes.
In the backward propagation stage, the adaptive threshold removes temporal dependencies between 5 [t). the gradient of the loss the gradient, enabling highly parallelizable training. Given I[t] = w w's'
with respect to the weight can be expressed as:
0s! it
0z
OC as as az
C
(KC (K+ S s
(15)
a
Ow
Sequential Training
Parallel Training
(1-) is the inner product. The derivative
as! [s) can be further defined as follows: a
a
C'g (C R(Z Vinlt) 0R 2
(16)
0z

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(-) denotes the surrogate gradient function. In this work, is implemented as a double Gaussian function 72). As shown Eq. [5| and Eq. [6] the gradient during backpropagation depends only on the current timestep. Consequently, the incorporation of the adaptive threshold introduces no additional training complexity, while encouraging sparse spike activity and preserving low computational cost. Detailed proof is provided in the Appendix.
B
5
Experiment
5.1 Compare with the SOTA methods
To validate the effectiveness of our proposed method, we conduct experiments on multiple time-series datasets. All experiments are conducted at least five times. First, we compare the performance of D-RF with other SOTA models commonly datasets, including Spiking Heidelberg Digits (SHD) [8] with 250 timesteps, Sequential MNIST, Permuted Sequential MNIST (S/PS-MNIST) [1 with 784 timesteps, and the more challenging Sequential CIFAR10 3-CIFAR10) 7 with 1024 timesteps. As presented Table I
D-RF achieves SOTA performance while using fewer or comparable parameters.
Table 1: Performance Comparison of Various Models.
Dataset
Method
Model Size
Type
Parallel
Dendritic
Acc.
LIF 79
85.1K
FF
x
x
72.06 10.00
ALIF 72
156.3K
Rec
X
x
98.70 94.30
BRF 21
68.9K
Rec
X
X
99.10 / 95.20
S/PS-MNIST
PSN 14
2.5M
FF
X
97.90 / 97.80
(784 Timesteps)
TC-LIF
155.1K
Rec
X
99.20 / 95.36
DH-LIF 80
0.8M
Rec
X
98.9 / 94.52
PMSN
156.4K
FF
99.50 97.80
Ours
155.1K
FF
99.50 98.20
LIF
249.0K
Rec
X
x
84.00
ALIF 72
141.3K
X
84.40
BRF 21
108.8K
Rec
X
92.50
SHD
PSN 14
232.5K
FF
89.75
(250 Timesteps)
TC-LIF 79
141.8K
Rec
X
88.91
DH-LIF 80
0.5M
Rec
X
91.34
PMSN
199.3K
FF
95.10
Ours
155.1K
FF
96.20
LIF 74
0.18M
FF
X
X
45.07
PSN 14
6.47M
55.24
S-CIFAR10
(1024 Timesteps)
SPSN
0.18M
FF
70.23
PMSN
0.21M
FF
82.14
Ours
0.21M
FF
84.30
Additionally, experimental results demonstrate that the RF-based model consistently outperforms the similarly sized LIF model across all datasets. It further confirms the temporal modeling capabilities of RF neurons in long sequence tasks. Moreover, our model also demonstrates superior recognition performance compared other dendritic models. Compared to the DH-LIF model [80), our approach enables parallel computation, effectively lowering the training speed associated with dendritic neurons. Compared the PMSN model [7], threshold resetting strategy prevents frequent spike generation and more effectively captures temporal dependencies. On the more challenging Sequential CIFAR10 dataset, our method achieves recognition accuracy of 84.20% representing 2.16% improvement.
We also evaluate our approach on the more challenging LRA benchmark (60] shown Table 2 model achieves significantly higher recognition accuracy than other neural models. Notably,

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achieves 60.02% accuracy on the ListOps task. It also demonstrates strong performance on tasks with longer timesteps, reaching 86.52% precision on the Text task (409 timesteps) and 90.02% precision on the Retrieval task (4000 timesteps). For the Image Task, D-RF attains 85.32% accuracy which underperforming SpikingSSM [5]]. This gap stems from the use of LayerNorm in SpikingSSMs, which reduces temporal variance. Furthermore, the performance gap between our model and ANN- based approaches like S4 [19 is no greater than 3%. Specifically, our model's accuracy is 0.42% higher for ListOps tasks. These findings highlight the strong temporal modeling ability of our method.
Table 2: Comparison of Model Accuracy on LRA Benchmark.
Model
ListOps (2,048)
Text
Retrieval (4,000)
Image (1,024)
Pathfinder (1,024)
SNN
Avg.
(Input length)
(4,096)
Random
10.00
50.00
50.00
10.00
50.00
34.00
Transformer 62
X
36.37
64.27
57.46
42.44
71.40
54.39
S4 (Bidirectional)
19
X
59.60
86.82
90.90
88.65
94.20
84.03
Binary S4D [57
54.80
82.50
85.03
82.00
79.79
77.39
+ GSU & GeLU
59.60
86.50
90.22
85.00
91.30
82.52
SpikingSSMs 51
60.23
80.41
88.77
88.21
93.51
82.23
Spiking LMU 33
37.30
65.80
79.76
55.65
72.68
62.23
ELM Neuron 566
44.55
75.40
84.93
49.62
71.15
69.25
SD-TCM 24
59.20
86.33
89.88
84.77
91.76
82.39
Ours
60.02
86.52
90.02
85.32
92.36
82.88
5.2
Sparser Spike with Accelerated Training
To evaluate the sparsity of the D-RF method, we compare the spike firing rates and theoretical energy
consumption 45) with those of similar approaches the LRA dataset [60]. As shown in Table 3 our model demonstrates a significant advantage. Specifically, on the ListOps task (39), achieves a Spiking Rate of 9.8% and energy consumption of 62. 22.4 Moreover, the Image tasks (30), the spike firing rate is reduced by 49.7% compared to the SD-TCM method. These results highlight the effectiveness of the adaptive threshold mechanism in enhancing energy efficiency.
(a) SD-TCM Method
(b) Our Method
Input
Input
U
Output
Output
20
40
60
80
100
0
20
40
60
80
100
Threshold
Membrance Potential
Continuous Spike
Sparse Spike
Figure 3: Comparison of Spiking Behavior Across Different Methods: (a) The SD-TCM method exhibits continuous spiking activity, as reflected in membrane potential and spike output. (b) The D-RF method shows sparser spike generation, indicating more efficient spiking behavior.
Additionally, we visualize the spiking behaviors of different reset methods to further the sparsity of our method. The SD-TCM model (24] assumes that a learnable decay constant can effectively approximate the reset process, with static threshold. In contrast, our method dynamically adjusts the threshold based on previous timesteps. As shown in Fig. 3 the proposed method significantly increases spike sparsity, indicating greater potential for energy efficiency.
Our method also ensures high training efficiency. We compare the per-epoch training cost across different sequence lengths. As shown in Fig. (a), we visualize the average epoch time under a batch size of 128. The results indicate that the proposed method achieves higher execution efficiency as the sequence length increases. At sequence length of 32768, our method achieves 581 x speedup over BPTT 68). Furthermore, compared to the SDN method [5I), our method also shows competitive performance, achieving 4.2x acceleration. We further validate the strategy Text tasks (4096). As

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Table 3: Comparison of Metrics across the LRA Benchmark.
Metric
Method
ListOps
Text
Refreival
Image
Pathfinder
Avg.
SpikingSSM [51]
13.2
10.1
6.9
22.1
7.4
11.9
Spiking Rate (%)
SD-TCM 24
11.2
7.9
5.7
15.7
5.8
9.3
Ours
9.8
6.3
3.3
7.9
3.2
SpikingSSM [51
84.2
355.2
237.0
708.9
65.1
290.1
Energy Cost (mJ)
SD-TCM 24
71.4
277.8
195.7
503.6
51.0
220.6
Ours
62.5
221.5
113.3
253.4
28.1
135.8
Results reproduced by ourselves, original code not publicly available.
shown in Fig. b). benefiting from highly parallelized membrane potential accumulation and spike generation processes, D-RF achieves faster simulation on GPU, yielding a 1.1 speedup. Moreover, the adaptive threshold mechanism enables the transformation of the serial accumulation-dest firing process into a parallelizable form, resulting in up to 147x x training acceleration. These results confirm that D-RF effectively addresses the high training cost.
Time (5)
(a) Comparison of Training Runtime
Time (9)
(b) Soq. vs Par. Training Cost
(c) Frequzney Rosponses of D-RF
10a
BPTT
SLTT
102
Sequential Parailel
148x
1.0
10a
SDN
Ours
581x
10
147x
8.6
10"
10"
Text (4096)
Brunch
4.2%
0.3
10"
10-1
1.1x
10-2
Lengsh
10"2
0.1
64
256
1024
4096
16384
32768
Forward
Buckward
Total
Frequency (Hz)
Figure 4: Train Speed and Frequency Analysis of D-RF: (a) Comparison of Training Runtime. (b) Sequential vs. Parallel Training Cost: Our method accelcrates both forward and backward
propagation. (c) Frequency Responses of D-RF: D-RF exhibits a broader frequency response.
5.3 Ablation Experiment
Table 4: Ablation Experiment
To assess the impact of the dendritic structure and the adaptive threshold mechanism, we conduct ablation exper- iments on the S-CIFARIO and ListOps datasets. We compare the performance different numbers of dendrites (n 4, 8, 16). As shown in Table 4 the model performance improves with an increas- ing number of dendrites. Considering the trade-off between complexity and performance, we set the number of dendrites to n = 4 for the for the LRA dataset. As shown in Fig. 3(c), visualize the fre-
Number of Dendrites
Dataset
Method
n=1
n=8
n=16
adaptive
80.3
84.3
84.6
85.1
S-CIFARIO
w/o
79.2
83.9
84.1
84.9
adaptive w/o 54.2
55.2
59.1
60.2
60.3
ListOps
58.9
59.2
59.6
S-CIFAR10 dataset and quency responses of individual dendritic branches. The results indicate that the proposed D-RF neuron captures nearly the entire frequency spectrum, further confirming the effectiveness of the dendritic structure. addition, we evaluate the effectiveness of the adaptive threshold mechanism. Experimental results demonstrate that the adaptive threshold improves model performance, primarily by effectively suppressing the adverse impact of redundant feature information on the results.
Conclusion
Inspired by the dendritic structure of biological neurons, this study proposes the D-RF model to further enhance the performance of SNNs on time-series signals. It consists of a multi-dendritic and soma structure. The multi-dendritic structure consists of branches with distinct decay factors, enabling the neuron to effectively extract soma with an adaptive threshold that ensures sparse spiking while enabling parallelizable computation. Extensive experiments demonstrate that our model achieves competitive results while maintaining sparse spiking activity for efficient training. These results highlight the strong potential of D-RF method to enable effective and efficient long sequence modeling edge-computing platforms.
information from the input signal. The

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7 Acknowledgments
This work is supported in part by the National Natural Science Foundation of China (No. 62220106008 and 6227 1432), in part by Sichuan Province Innovative Talent Funding Project for Post- doctoral Fellows (BX202405), part by the Shenzhen Science and Technology Program (Shenzhen Key Laboratory. Grant No. Introducing Innovative and Entrepreneurial Teams, Grant No. 2023ZT10X042 and in part by the State Key Laboratory of Brain Cognition and Brain-inspired Intelligence Technology, Grant No. SKIBB-KK025010.
in part by the Program for Guangdong
References
[1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv preprint arXiv:2303.08774, 2023.
[2] Angel Alonso and Rodolfo R Llinás. Subthreshold na+-dependent theta-like rhythmicity in stellate cells of entorhinal cortex layer ii. Nature, 342(6246):155-177, 1989.
[3] Kendall E Atkinson. An introduction to numerical analysis. John wiley & sons, 2008.
[4] Guillaume Bellec, Darjan Salaj, Anand Subramoney, Robert Legenstein, and Wolfgang Maass. Long short-term memory and learning-tor cooning-to-leam in networks spiking neurons. Advances in neural information processing systems, 31, 2018.
[5] Siqi Cai, Zheyuan Lin, Xiaoli Liu, Wenjie Wei, Shuai Wang, Malu Zhang, Tanja Schultz, and Haizhou Li. Spiking neural networks for eeg signal analysis: From theory to practice. Neural Networks, page 1081 17,1 2025.
[6] Stefano Caviglia, Maurizio Valle, and Chiara Bartolozzi. Asynchronous, event-driven readout of posfet devices for tactile sensing. In 2014 IEEE International Symposium on Circuits and Systems (ISCAS), pages 2648-2651. IEEE, 2014. doi: (011199/scaa.221..6888
[7] Xinyi Chen, Jibin Wu, Chenxiang Ma, Yinsong Yan, Yujie Wu, and Kay Chen Tan. Pmsn: A parallel multi-commartment spiking neuron for multi-scale temporal processing, arXiv preprint arXiv:2408.1 14917, 2024.
[8] Benjamin Cramer, Yannik Stradmann, Johannes Schemmel, and Friedemann Zenke. The heidelberg spiking data sets for the systematic evaluation of spiking neural networks. IEEE Transactions Neural Networks and Learning Systems, 33(7):2744-2757) 2020.
[9] Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yongqiang Cao, Sri Harsha Choday, Georgios Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, et al. Loihi: A neuromorphic manycore processor with on-chip learning. leee Micro, 38(1):82-99, 2018.
[10] Raymond DeCarlo. Linear systems: A state variable approach with numerical implementation. Prentice-Hall, Inc., 1989.
[11] Shikuang Deng, Yuhang Li, Shanghang Zhang, and Shi Gu. Temporal efficient training of spiking neural network via gradient re-weighting. arXiv preprini arXiv:2202.1 1996, 2022.
[12] Peter U Diehl, Daniel Neil, Jonathan Binas, Matthew Cook, Shih-Chii Liu, and Michael Pfeiffer. Fast-classifying, high-accuracy spiking deep networks through weight and threshold balancing. In 2015 International joint conference on neural networks (IJCNN), pages 1-8. ieee, 2015.
[13] Wei Fang, Zhaofei Yu, Yanqi Chen, Timothée Masquelier, Tiejun Huang, and Yonghong Tian. Incorporating learnable membrane time constant enhance learning of spiking neural networks. In Proceedings of the IEEE/CVF international conference on computer vision, pages (671,,
2021.
[14] Wei Fang. Zhaofei Yu, Zhaokun Zhou, Ding Chen, Yanqi Chen, Zhengyu Ma, Timothée Masquelier, and Yonghong Tian. Parallel spiking neurons with high efficiency and ability to learn long- mergttermee dependencies. Advances in Neural Information Processing Systems, 36, 2024.
10

---

[15] E Paxon Frady, Sophia Sanborn, Sumit Bam Shrestha, Daniel Ben Dayan Rubin, Garrick Orchard, Friedrich T Sommer, and Mike Davies. Efficient neuromorphic signal processing with resonator neurons. Journal of Signal Processing Systems, 94(10):917-927, 2022.
[16] Chittotosh Ganguly, Sai Sukruth Bezugam, Elisabeth Abs, Melika Payvand, Sounak Dey, and Manan Suri. Spike frequency adaptation: bridging neural models and neuromorphic applications. Communications Engineering, 3(1):22 2024.
[17] Wulfram Gerstner and Werner M Kistler. Spiking neuron models: Single neurons, populations, plasticity. Cambridge university press, 2002.
[18] Albert Gu, Karan Goel, Ankit Gupta, and Christopher Ré. On the parameterization and initialization of diagonal state space models. Advances in Neural Information Processing Systems, 35:35771-3599933 2022.
[19] Albert Gu, Karan Goel, and Christopher Ré. Efficiently modeling long sequences with structured state spaces. arXiv preprint arXiv:2111.00330 2021.
[20] Albert Gu, Isys Johnson, Karan Goel, Khaled Saab, Tri Dao, Atri Rudra, and Christopher Ré. Combining recurrent, convolutional, and continuous-time models with linear state space layers. Advances in neural information processing systems, 34:572-585, 2021.
[21] Saya Higuchi, Sebastian Kairat, Sander Bohte, and Sebastian Otte. Balanced resonate-and-fff neurons. In Forty-first International Conference on Machine Learning.
[22] Julian Hille, Daniel Auge, Cyprian Grassmann, and Alois Knoll. Resonate-and-Fin neurons for radar interference detection. In Proceedings of the International Conference Neuromorphic Systems 2022, pages 1-4, 2022.
[23] JiaKui Hu, Man Yao, Xuerui Qiu, Yuhong Chou, Yuxuan Cai, Ning Qiao, Yonghong Tian, Bo Xu, and Guoqi Li. High-performance temporal reversible spiking neural networks with o(l) training memory and o(1) inference cost. arXiv preprint arXiv:2405. 16466, 2024.
[24] Yulong Huang, Zunchang Liu, Changchun Feng, Xiaopeng Lin, Hongwei Ren, Haotian Fu, Yue Zhou, Hong Xing, and Bojun Cheng. Prf: Parallel resonate and fire neuron for long sequence learning in spiking neural networks. arXiv preprint arXiv:2410.03530, 2024.
[25] Binyuan Hui, Jian Yang, Zeyu Cui, Jiaxi Yang, Dayiheng Liu, Lei Zhang, Tianyu Liu, Jia- jun Zhang, Bowen Yu, Keming Lu, et al. Qwen2. S-coder technical report. arXiv preprint arXiv:2409. 12186, 2024.
[26] Ana R Inácio, Ka Chun Lam, Yuan Zhao, Francisco Pereira, Charles R Gerfen, and Soohyun Lee. wrain-wide presynaptic networks of functionally distinct cortical neurons. Nature, pages 1-11, 2025.
[27] Eugene M Izhikevich. Resonate-and-fit neurons. Neural networks, 14(6-7):883-844 2001.
[28] Eugene M Izhikevich. Simple model of spiking neurons. IEEE Transactions on neural networks, 1466:1566- 2003.
[29] Mikhail V Koroteev. Bert: review of applications in natural language processing and under- standing. arXiv preprint arXiv:2103.1194 2021.
[30] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[31] Quoe V Le. Navdeep Jaitly, and Geoffrey E Hinton. A simple way to initialize recurrent networks rectified linear units. arXiv preprint arXiv: 00941, 2015.
[32] Drew Linsley, Junkyung Kim, Vijay Veerabadran, Charles Windolf, and Thomas Serre. Learning long-range spatial dependencies with horizontal gated recurrent units. Advances in neural information processing systems, 31, 2018
[33] Zeyu Liu, Gourav Datta, Anni Li, and Peter Anthony Beerel. Lmuformer: low complexity yet powerful spiking model with legendre memory units. arXiv preprint 24X::2402.048822 2024.

---

[34] Rodolfo R Llinas, Anthony A Grace, and Yosef Yarom. In vitro neurons in mammalian cortical layer exhibit intrinsic oscillatory activity in the 10-to 50-hz frequency range. Proceedings of the National Academy of Sciences, 88(3):897-901, 1991.
[35] Yi Luo, Zhuo Chen, and Takuya Yoshioka. Dual-path rnn: efficient long sequence modeling for time-domain single-channel speech separation. In ICASSP 2020-2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 46-50. IEEE, 2020.
[36] Andrew Maas, Raymond E Daly, Peter T Pham, Dan Huang, Andrew Y Ng. and Christopher Potts. Learning word vectors for sentiment analysis. In Proceedings of the 49th annual meeting of the association for computational linguistics: Human language technologies, pages 142-150,
2011.
[37] Wolfgang Maass. Networks of spiking neurons: the third generation of neural network models. Neural networks, 109961659-1671, 1997.
[38] Simon Musall, Xiaonan R Sun, Hemanth Mohan, Xu An, Steven Gluf, Shu-Jing Li, Rhonda Drewes, Emma Cravo, Irene Lenzi, Chaoqun Yin, et al. Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making. Nature neuroscience, 26(3):495-505, 2023.
[39] Nikita Nangia and Samuel R Bowman. Listops: A diagnostic dataset for latent tree learning. arXiv preprint arXiv: 1004.06040005 2018
[40] Emre O Neftei, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient learning in spiking neural networks:
Bringing the power of gradient-based optimization to spiking neural networks. IEEE Signal Processing Magazine, 36(6):51-66) 2019.
[41] Garrick Orchard, E Paxon Frady, Daniel Ben Dayan Rubin, Sophia Sanbor, Sumit Bam Shrestha, Friedrich T Sommer, and Mike Davies. Efficient neuromorphic signal processing with loihi 2. In 2021 IEEE Workshop on Signal Processing Systems (SiPS), pages 254-259. IEEE, 2021.
[42] Christine Pedroarena and Rodolfo Llinás. Dendritic calcium conductances generate high- frequency oscillation in thalamocortical neurons. Proceedings of the National Academy of Sciences, 94(2):724-728, 1997.
[43] Dragomir R Radev, Pradeep Muthukrishnan, Vahed Qazvinian, and Amjad Abu-Jbara. The acl anthology network corpus. Language Resources and Evaluation, 47:919 944, 2013.
[44] Wilfrid Rall. Theoretical significance of dendritic trees for neuronal input-output relations (1964). 1994.
[45] Nitin Rathi and Kaushik Roy. Dict-snn: Direct input encoding with leakage and threshold optimization in deep spiking neural networks. arXiv preprint arXiv:2008.0366 036588 2020.
[46] James P Roach, Aleksandra Pidde, Eitan Katz, Jiaxing Wu, Nicolette Ognjanovski, Sara Aton, and Michal R Zochowski. Resonance with subthreshold oscillatory drive organizes activity and optimizes learning in neural networks. Proceedings of the National Academy of Sciences, 1 115(13)E3011--3302 2018.
[47] Bodo Rueckauer, AlexanAae Lungu, Yuhuang Hu, Michael Pfeiffer, and Shih-Chin Liu. Conversion of continuous- -valuud classification. Frontiers in neuroscience. 11:682, 2017.
valued deep networks efficient event-driven networks for image
[48] Md Abu Sayeed, Saraju P Mohanty, Elias Kougianos, and Hitten P Zaveri. eseiz: An edge- device for accurate seizure detection for smart healthcare. IEEE Transactions on Consumer Electronics, 63(3):377-387, 2019.
[49] Mike Schuster and Kuldip K Paliwal. Bidirectional recurrent neural networks. IEEE transactions on Signal Processing, 45(1 ():211)::332211 26811 1997.
[50] Ahmed Shaban, Sai Sukruth Bezugam, and Manan Suri. An adaptive threshold neuron for recurrent spiking neural networks with nanodevice hardware implementation. Nature Communi- cations, 12(1):4234, 1021.
12

---

[51] Shuaijie Shen, Chao Wang, Renzhuo Huang, Yan Zhong, Qinghai Guo, Zhichao Lu, Jianguo Zhang, and Luziwei Leng. Spikingssms: Learning long sequences with sparse and parallel spiking state space models. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 20380-20388, 2025.
[52] Alex Sherstinsky. Fundamentals of recurrent neural network (rnn) and long short-term memory (Istm) network. Physica D: Nonlinear Phenomena, 404: 132306, 2020.
[53] Sumit Bam Shrestha, Jonathan Timcheck, Paxon Frady, Leobardo Campos-Macias and Mike Davies. Efficient video and audio processing with loihi 2. In ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 13481- 13485. IEEE, 2024.
[54] TSAN Simões, CIN Sampaio Filho, HJ Herrmann, JS Andrade Jr, and L de Arcangelis. Ther- modynamic analog of integrate-and and-fire neuronal networks by maximum entropy modelling. Scientific Reports, 14(1):9480, 2024.
[55] Jimmy TH Smith, Andrew Warrington, and Scott W Linderman. Simplified state space layers for sequence modeling, arXiv preprint ar0iv: 200.:2208.0433,, 2022.
[56] Aaron Spieler, Nasim Rahaman, Georg Martius, Bernhard Schölkopf, and Anna Levina. The expressive leaky memory neuron: an efficient and expressive phenomenological neuron model can solve long-horizon tasks. arXiv preprint arXiv:2306. 16922, 2023.
[57] Matei-loan Stan and Oliver Rhodes. Learning long sequences in spiking neural networks. Scientific Reports, 14(1):21957, 2024.
[58] Qiaoyi Su, Shijie Mei, Xingrun Xing, Man Yao, Jiajun Zhang, Bo Xu, and Guoqi Li. Snn- bert: training-efffiienn spiking neural networks for energy-efficient bert. Neural Networks, 180: 106630, 2024.
[59] Amirhossein Tavanaei, Masoud Ghodrati, Saeed Reza Kheradpisheh, Timothée Masquelier, and Anthony Maida. Deep learning in spiking neural networks. Neural networks, 111:47-63, 2019.
[60] Yi Tay, Mostafa Dehghani, Samira Abnar, Yikang Shen, Dara Bahri, Philip Pham, Jinfeng Rao, Liu Yang. Sebastian Ruder, and Donald Metzler. Long range arena: A benchmark for efficient transformers. arXiv preprint wrXiv:2011.04066 2020.
[61] Roger D Traub, Robert K Wong. Richard Miles, and Hillary Michelson. A model of ca3 hippocampal pyramidal neuron incorporating voltage-clamp data on intrinsic conductances. Journal of neurophysiology, 66(2):635-65) 1991.
[62] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. Advances in neural information processing systems, 30, 2017.
[63] Jingya Wang, Xin Deng, Wenjie Wei, Dehao Zhang, Shuai Wang, Qian Sun, Jieyuan Zhang, Hanwen Liu, Ning Xie, and Malu Zhang. Training-free ann-to-snn conversion for high- performance spiking transformer. arXiv preprint arXiv:2508.0777 07710, 2025.
[64] Shuai Wang, Dehao Zhang, Ammar Belatreche, Yichen Xiao, Hongyu Qing, Wenjie Wei, Malu Zhang, and Yang Yang. Ternary spike-based neuromorphic signal processing system. Neural Networks, 187:10 2025.
[65] Shuai Wang, Dehao Zhang, Kexin Shi, Yuchen Wang, Wenjie Wei, Jibin Wu, and Malu Zhang. Global-local convolution with spiking neural networks for energy-efficient keyword spotting. arXiv preprini arXiv:2406.14 13179, 2024.
[66] Shuai Wang, Malu Zhang. Dehao Zhang, Ammar Belatreche, Yichen Xiao, Yu Liang, Yimeng Shan, Qian Sun, Enqi Zhang, and Yang Yang. Spiking vision transformer with saccadic attention. arXiv preprint
2025.
13

---

[67] Xiaying Wang, Michael Hersche, Batuhan Tömekce, Burak Kaya. Michele Magno, and Luca Benini. An accurate eegnet-based motor-imagery brain- computer interface for low- -ewpoowe edge computing. In 2020 IEEE international symposium on medical measurements and applications (MeMeA), pages 1-6. IEEE, 2020.
[68] Yujie Wu, Lei Deng, Guoqi Li, Jun Zhu, and Luping Shi. training seghhpeeformance spiking neural networks. Frontiers in neuroscience, 12:331, 2018.
backpropagation for
[69] Yichen Xiao, Shuai Wang, Dehao Zhang. Wenjie Wei, Yimeng Shan, Xiaoli Liu, Yulin Jiang, and Malu Zhang. Rethinking spiking self-attention mechanism: Implementing a-xnor similarity calculation in spiking transformers. In Proceedings of the Computer Vision and Putlern Recognition Conference, pages 5444-5454, 2025.
[70] Yinsong Yan, Qu Yang. Yujie Wu, Hanwen Liu, Malu Zhang, Haizhou Li, Kay Chen Tan, and Jibin Wu. Efficient and robust temporal processing with neural oscillations modulated spiking neural networks. Nature communications, 16(1):8651, 2025.
[71] An Yang, Anfeng Li, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chang Gao, Chengen Huang, Chenxu Lv, et al. Qwen3 technical report. arXiv preprint arXiv:2505.09388 2025.
[72] Bojian Yin, Federico Corradi, and Sander M Bohté. Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks. Nature Machine Intelligence, ((10):905-913) 2021.
[73] Huifeng Yin, Hanle Zheng, Jiayi Mao, Siyuan Ding. Xing Liu, Mingkun Xu, Yifan Hu, Jing Pei and Lei Deng. Understanding the functional roles of modelling components in spiking neural networks. Neuromorphic Computing and Engineering. 4(3):034009, 2024.
[74] Friedemann Zenke and Tim P Vogels. The remarkable robustness of surrogate gradient learning for instilling complex function in spiking neural networks. Neural computation, 33(4):899-925,
2021.
[75] Dehao Zhang, Shuai Wang. Ammar Belatreche, Wenjie Wei, Yichen Xiao, Haorui Zheng, Zijian Zhou, Malu Zhang, and Yang Yang. Spike-based neuromorphic model for sound source localization. Advances in Neural Information Processing Systems, 37:1 37:1139114 1399111119366.111111111111111111 2024.
[76] Dehao Zhang, Shuai Wang, Yichen Xiao, Wenjie Wei, Yimeng Shan, Malu Zhang, and Yang Yang. Memory-free and parallel computation for quantized spiking neural networks. In ICASSP 2025-2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1-5, IEEE, 2025.
[77] Malu Zhang. Jiadong Wang, Jibin Wu, Ammar Belatreche, Burin Amonpaisannnn, Zhixuan Zhang, Venkata Pavan Kumar Miriyala, Hong Qu, Yansong Chua, Trevor E Carlson, et al. Rectified linear postsynaptic potential function for backpropagation in deep spiking neural networks. IEEE transactions neural networks and learning systems, 33(5):1947-199) 2021.
[78] Malu Zhang, Shuai Wang, Jibin Wu, Wenjie Wei, Dehao Zhang, Zijian Zhou, Siying Wang, Fan Zhang, and Yang Yang. Toward energy energy-efficien efficient spike- spike-based deep reinforcement learning with temporal coding. IEEE Computational Intelligence Magazine, 20(2):45 57, 2025.
[79] Shimin Zhang, Qu Yang, Chenxiang Ma, Jibin Wu, Haizhou Li, and Kay Chen Tan. Te-lif: A two-compartment spiking neuron model for long-term sequential modelling. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 16838-16847, 2024.
[80] Hanle Zheng, Zhong Zheng, Rui Hu, Bo Xiao, Yujie Wu, Fangwen Yu, Xue Liu, Guoqi Li, and Lei Deng. Temporal dendritic heterogeneity incorporated with spiking neural networks for learning multi-timescale dynamics. Nature Communications, 15(1):277, 2024.
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21

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A Detail of Resonate-and-Fire Neuron
A.1 Dynamics of Resonate-and-Fire Neuron
The RF neuron process can be modeled as a first-order linear oscillatory system:
d
di =1) +b+ it) I(),,
(I)
where z(t) = u(t) + it(t) E C represents the complex state of the neuron. b < 0 is the damping factor, is the angular frequency, and Z(t) the external input signal the neuron.
A.2 Discretization via Exponential Euler Method
We use the exponential Euler method to discretize the continuous differential equation as follows:
ztt + 6) exp {(b + im) x(t) + 246
exp + (t + 5 (s)]
(2)
where å is discrete the timesteps. Assuming that the input I(s) remains approximately constant over the small interval [t, + 87. This discretization can be derived as follows:
2(t 6) = cx +
exp((b + it) + 5 - -s)}I(s)ds
146
exp((b+iw)0]x +I())
exp((b+iw)) + s)}ds
exp (t) - $s} 5=
exp((b+i<)0] + z(t)
6 iw
-1
-1
+ z(t)
b + iw e
expl + iw)o})
b+iw
exp + iw))+ + I(t) 1- exp((( ((b iw)d} {b))
b iw
When d + 0, the second term can be represented using Taylor expansion
I(t) exp + iw)o} + iw
8I(() O
(4)
Therefore, the discretized form of Eq. be expressed as:
2 f = exp + ( - 1] + 8T
(5)
=|{] represents the complex state at discrete timestep 8 is the time interval, and Z[[] is the external input at timestep t.
A.3 Frequency Band Preference Characteristics
To examine the RF neuron's 's response to the periodic input, we seek particular solution to Eq. with the same frequency as I(t):
I = Io (xp(1 =[(] = exp(iinn))
(6)
where H is complex constant that determines both amplitude and the phase of the response. It can be rewritten as follows:
H exp exp(6(6 + iw) ÓA.
Therefore, the H can be defined as follows:
5A
8A exp(is25) (1 exp( + in expl exp(-i26)))
(8)
- + b0)}
The magnitude of the transfer function is:
8
H (exp(())))
1 -eep
(bb + i(8w - 22)
22

---

Therefore, single RF neuron can be regarded as first-order band-pass filter with resonance peak 52 =s w, and a narrow frequency band determined by the damping factor db.
For the phase diagram of RF neuron, the phase shift between input and response provides additional information about resonance properties:
o( = arg (1 tan -1 ( w
(10)
At resonance Ω w, the phase shift -90%. Therefore, when resonance occurs, the phase of RF neurons will accumulate rapidly.
A.4
Reset Mechanism in Resonate-and-Fing -ire Neuron
A.4.1
Traditional Soft and Hard Reset
The traditional soft and hard reset mechanisms for the RF neuron can be defined as follows:
0. S=(4] = [(=1{} - Ven. if (= > Vih. soft reset,
if (2(e)} > Vin. hard reset
(11)
As shown in Eq. 11, when the imaginary part of the RF neuron's 's state exceeds the threshold, a spike is generated. For the hard reset process, the imaginary part of the RF neuron will be reset to O. For the soft reset process, the imaginary part of the RF neuron will be subtracted from the threshold. Both of them disrupt the original oscillatory dynamics of the RF neuron.
Under reset conditions, the system becomes nonlinear, preventing direct application of the Z-ttransform. However, we can model the reset mechanism nonlinear perturbation term:
# = exp( + iw) zR| - 1| + Z[[] + d[t],
(12)
where d[{ represents the perturbation introduced by the reset operation:
d(4 Σ (=R[/a] - =[(x]) 8(1 [x],
(13)
with 8/t - [x] being the unit impulse time 4.. We applying the 7-tarssfoomm yields:
- + [(d
Z(zn[t))
(14)
exp (b- + iw))==1
For periodic reset patterns, the frequency response contains a series of harmonic components:
Hn(w') H(w') + Σ Cn . -606 - w miyr),
(15)
11=-00
where (w") 1-e(0-t))- represents the frequency response of the original system, On the Fourier coefficients, and w, is the reset frequency. This demonstrates that the reset mechanism introduces sideband components in the frequency domain, thereby weakening the frequency selectivity of RF neurons. Therefore, conventional reset mechanisms inevitably compromise the -electiviy property of RF neurons to some extent.
A.4.2 No Reset Mechanism
In the absence of reset mechanism, the band-selectivity of RF neurons can be effectively preserved; however, it will lead to frequent spike emissions.
Consider sinusoidal input at the resonant frequency: I[I] In exp(iwt). The solution to the difference equation can be obtained by recursive expansion:
z|t| = exp {bb + iw)t 2|0 + Io
Σ
exp (bb + (tw) - 1)} exp {iwk}
A=
(16)
1 exp
= exp (bb} 2 + Io exp [itt]
1- exp
23

---

When < 0 and & >>> 0, the first term vanishes, yielding:
To exp(iwt)
To (in(t))
< R
Im(2[t)
(17)
1 - exp(b}}
1 - -xxp(b}
Therefore, in long sequence tasks, RF neurons are more likely to exhibit frequent spike emissions. The duration of each spike given by:
1 -- 2 Vin (Vin(1-exp(0)) exp(b)) A sppik
(18)
To
where Ven is the threshold. The maximum spike duration corresponds to half of the oscillation cycle. However, in the absence of reset mechanism, the time-invariant parameter design allows the RF neuron to be interpreted as a convolution between a fixed kernel IC and the input Z[t]_ This structure offers clear advantage during training, as it requires only ((C) computational complexity.
A.4.3 Adaptive Decay Factor for RF Neuron
To balance the frequency band selectivity of RF neurons and reduce their firing rates, Higuchi et al. [2 introduce refractory mechanism into RF neurons, effectively suppressing excessive spike emissions. This mechanism consists of two main components:
(in 6. + (t), b(t) b q(t).
(19)
where 0g is constant threshold and be is the decay factor. q(t) is the refractory period, which decays exponentially with time:
(t) = 70(t - 1) + s(t s(t-1]
(20)
Here, 7 = 0.9 is the default period constant. We further analyze the impact of the time-varying design of b(t) in conjunction with the frequency response, showing that it does not compromise the frequency band selectivity of RF neurons. As shown in Eq. 6-9, the frequency response of the BRF neuron at time can be defined as:
H = 1 - exp( (b(t) + it - -
(21)
This represents the system's 's response strength at time an input signal with frequency w', We can verify that the expression attains its maximum at w' = w by computing the derivative of this part. By taking the derivative of w, solve the extreme point situation of Eq. 21:
d (t,(''')) k22 Ow'
i
Ow - exx(((bb(() cos(w - =²))²² + (exp(b(1)) sin(w - w))) (22)
where K- = H((tw")) For the latter term, we can further simplify.
O|H(t,w'))
- 1
k3
Ow' [[1 - exp(b(t)) cos(w - w²))²² + (exp(b(t)) sin(w - w'))
Ow
1
33
Ow 2exx((b(t)) cos(w w') exp(2b(())]
(23)
2 b2 [2exp(b(()} sin(w - ww))
Therefore, for any b, the first-order derivative of H (t, w²)l at w' = w is zero, which is a stationary point. Therefore, we need further derive it and find that the derivative value of the second-order
derivative is:
8 (t,w')) Ou
=ttH(t,w)) (()) < O.
(24)
Therefore, w is the maximum point. When w' the band response is at its maximum value. Additionally, when spike emitted, the value b(t) will decrease, resulting in a faster decay rate.
For RF neurons, the absence of a reset mechanism leads to frequent spike emissions, while simply subtracting the threshold compromises their frequency-selective properties. Although Huchigu et al. [21] preserve both energy efficiency and frequency selectivity in RF neurons, their time-invariant factor significantly hinders model training. Therefore, designing an effective and principled reset mechanism is crucial for fully exploiting the potential of RF neurons.
24

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B Computational Complexity of Backpropagation in the D-RF Model
B.1 Gradient Calculation for Backpropagation
To make the backpropagation process clearer, we begin with the gradient of the loss function with respect the weight w', expressed as:
ac
a 1
[z
wi
(25)
1
Ow
Since ['[ is derived from the membrane potential [ through nonlinear activation function, need to compute the derivative of the spike signal with respect to the membrane potential. In this work, it is implemented as double Gaussian function 72. Therefore, this derivative is given by:
as iz
2AR 2
C'g (C [[Z() Villt)
(26)
dz [t]
where ((-) is the surrogate gradient function, C¹ is the coefficient for the current layer, and Vin[4] is the adaptive threshold. The membrane potential ['(t] is the weighted sum of the previous layer's spike signals [t]. Therefore, the derivative of the membrane potential with the weight w is:
az Ow Ii exp (D) St-1(t k].
(27)
where Si kt] represents the spike signal from the previous layer, I; is a constant, and dD is a parameter associated with dendritic branches. Substituting the above expressions into the gradient formula, we can obtain the following:
OC
d'g (C'R(Z(1) (i) Trexppk: (Irexpll 8D3 - (--1(-bb1) k1).
(28)
1
t=0
For faster training, transform the gradient computation into a parallel computation format using convolution. Thus, the final expression becomes:
Oc
as [1] [
(K * 5K* 11 111)
L
(29)
as t
where ( represents the inner product, and (K * st represents the convolution operation. The final gradient expression can be summarized as;
Oc
( 1
( Dw
Oc
a i (K
W C
(30)
15
This process demonstrates how the gradient calculation proceeds from the error term to the final weight update rule, incorporating both sequential and parallel training strategies for efficiency.
B.2 Analysis of Computational Complexity
As shown in Eq. 30, our method converts gradient computation into convolution operations, effec- tively leveraging the efficiency of the Fast Fourier Transform (FFT). This results in convolution computation time of ( log C). Since the FFT of length can be computed in O log L), and convolution is equivalent to multiplication in the frequency domain, the complexity of convolution reduces from ( in sequential training to OC L) in parallel training. Therefore, using the convolution-basl based parallel training approach ensures that the overall computational complexity of the backpropagation process is reduced to Oo C). making it significantly more efficient.
This analysis demonstrates that by utilizing parallel computation and FFT-based convolution, we can significantly reduce training costs, lowering the complexity from (C²)) to O(Clog L), which critical for long sequence modeling tasks.
25

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C Experiment Detail
C.1 Dataset Description
We provide more context and details for (P)S-MNIST and the tasks of the LRA 660 The following descriptions primarily reference [55].
Sequential MNIST (S (S-MNIST) 31 Each 28 x 28 grayscale MNIST image is reshaped into a one-dimensional sequence of 784 scalar values. The model must assign one of ten digit labels (0-9) based solely on this temporal input.
Permuted Sequential MNIST (PS-MNIST) 31: Follows the same flattening procedure as sMNIST to produce a 784-length sequence, which is then reordered by a fixed permutation before classification into digits 0-9.
Spiking Heidelberg Digits (SHD) [8] Consists of 10 000 spike-encoded patterns represent- ing handwritten digits (0-9) from the Heidelberg Digits dataset. Each sample is provided as either binary continuous spike trains for classification.
Sequential C-FAR-10 (S (S-CIFARIO) 7 14: Converts each 32 x 32 color image into sequence of 1 024 RGB triplets. The task is to classify the image into one of ten categories based on this sequential representation.
ListOps 39|:
dataset for evaluating sequence- based models. It contains mathematical operations (e.g., min, max) and integer operands ranging from 0-9, represented using prefix notation and brackets. The task is to compute the result of a mathematical expression (e.g.. [max 2 6 [min 9 7] 0] - 7). The characters encoded one-hot vectors and the sequences are padded to a maximum length of 2,000. There are 10 classes representing the results of the expression.
Text [6]]: A sentiment classification dataset based on the iMDB movie review dataset. The task is to classify given movie review as positive or negative. The characters are encoded as integer tokens and the sequences are padded to a maximum length of 4,096. There are two classes representing positive and negative sentiment. The dataset includes 25,000 training samples and 25,000 test samples.
Retrieval 43 A dataset based on the ACL Anthology Network corpus. The task is to classify whether text citations equal or not. Each eitation is encoded a sequence of integer tokens. References is compressed separately and then passed to the final classification layer. This dataset evaluates the model's ability to represent and retrieve textual relations.
An image classification dataset based on the CIFAR-10 10 dataset. contains 32x32 grayscale images that are flattened into sequence of length 1,024. The task is to classify each image into one of ten categories. The dataset contains 45,000 training samples, 5,000 validation samples, and 10,000 test samples.
Image 30
Pathfinder [32] A dataset for path finding tasks. It contains 32x32 grayscale images, each showing a start and end point represented as small circles. The task is to classify whether there is a dashed line (or path) connecting the start and end points. The sequences are padded to a length of 1, 1,024. The dataset includes 160,000 training samples, 20,000 validation samples, and 20,000 test samples. The data were normalized to the range [-1,1]
C.2 Model Architecture and Hyperparameter Setting
In this task, utilize network architecture composed of stacked residual blocks (RB) between membrane potentials. Each RB block comprises residual connection and a sequence of "D-RF Model - 1x1 convolution Spiking Neuron model Ixl convolution". Except for the possible floating-point multiplications involved in the neuron dynamics, all other operations are spike-driven, conusre-conttr edge devices. All experiments conducted on Ubuntu server equipped with NVIDIA GeForce RTX 4090 (24G Memory), Intel(R) Xeon(R) Platinum 8370C SPU@2.80GHH, PyTorch 2.1.0, and CUDA 11.8.
which is more favorable for deployment
Furthermore, we conducted a parameter count comparison between our model and baseline methods on the LRA dataset, further underscoring the efficiency of our approach. The learning rate for
26

---

Table Hyperparameters for LRA Task
Task
Depth
Norm
Channels
Pre-norm
Dropout
B
Epochs
Weight Decay
ListOps
8
BatchNorm
128
False
0
50
40
0.05
Text
6
BatchNorm
True
0
16
32
0.05
Retrieval
6
BatchNorm
256
True
0
32
20
0.05
Image
6
BatchNorm
512
False
0.1
50
200
0.05
Pathfinder
6
BatchNorm
256
True
0.05
64
200
0.03
neuron-specific parameters is set 0.001, while global learning rate of 0.005 applied the entire
network. Detailed hyperparameter settings are summarized in Table 5
As shown in Table 6 we report the model sizes across various LRA tasks, demonstrating that our approach maintains parameter efficiency comparable to other baselines. When combined with the performance metrics in Table 2 and the energy consumption results in Table 3 it becomes evident that our model not only achieves competitive accuracy but also induces significantly sparser spiking activity. These results underscore its potential modeling on edge platforms.
effective and efficient solution for long sequence
Table 6: Comparison of Model Size between Competing Networks across Tasks.
Metric
Method
S-CIFAR10
ListOps
Text
Retreival
Image
Pathfinder
$4
19
308K
815K
843K
3.6M
3.6M
1.3M
Parm.
SpikingSSM [51
308K
815K
843K
3.6M
3.6M
1.3M
SD-TCM [24]
272K
830K
1.1M
4.1M
1.3M
Ours
216K
297K
841K
1.1M
3.2M
1.3M
Results reproduced by ourselves, as the original code is not publicly available.
Additionally, we further report the model parameter configurations corresponding to different dendritic branches in the ablation study. While increasing the number of dendrites. leads to larger model and generally improves recognition performance, the gains tend to saturate. Specifically, on the image classification task, the model achieves an accuracy of 86.2% with 4 dendrites, and only marginal increase to 86.7% when the number is doubled to 8. Given the limited performance gain relative to the parameter overhead, we select n = as the default configuration for the S-CIFAR task.
Table 7: Ablation Experiment
Number of Dendrites
Dataset
Metric
Length
n=1
n=2
n=4
n=6
n=8
n=16
Accuracy (%)
1024 1024
80.3
82.1
84.3
84.4
84.6
85.1
S-CIFAR10
Parameter (M)
209K
213K
216K
219K
222K
234K
D Further Discussion & Limitation
Similar to Transformer architectures, D-RF neurons can also function as stackable computational components. They have demonstrated strong performance across various sequential benchmarks, such as LRA benchmark and S-CIFARIO dataset. However, the sequence lengths of these datasets remain relatively limited compared with those used in current large language models (LLMs). Recent ANN-based L.MSs can process input sequences ranging from 16K to 1M tokens 25 71, but such capabilities typically rely on complex architectural designs and large parameter scales, leading to substantial computational overhead. In this paper, we adapt neural dynamics perspective, aiming to enhance memory capacity and temporal information processing.
Further Discussion:
This study has two limitations. First, DRF neurons improve model performance on long sequence tasks through efficient computation. However, the additional MAC operations they introduce and their reliance on complex-valued operations may restrict their deployment on neuromorphic chips. Second, the study primarily focused on classification tasks and does not address regression tasks, such as text generation and long sequence prediction. Therefore, future work will extend to text generation tasks and further explore implementation strategies on neuromorphic chips.
Limitation:
27


Reviews:
[{"review_id": "S0iqhXRfSt", "reviewer": "Reviewer_2jgg", "summary": "This paper proposes D-RF neurons to address the limitations of RF neurons in long sequence modeling. The work provides comprehensive theoretical analysis, revealing RF neurons' limited frequency band perception and energy-performance trade-offs through frequency domain analysis. The method is well-designed with D-RF employing multi-dendritic architecture for full-spectrum coverage and adaptive threshold mechanisms that balance training efficiency with sparse spike generation.", "questions": "1.To validate D-RF neurons' temporal processing capabilities, evaluation on speech datasets and neuromorphic datasets would be necessary.  \n2. Given the complex neural dynamics of D-RF neurons, providing pseudocode or algorithmic flowcharts would be beneficial for understanding and reproducibility.  \n3. The authors are encouraged to provide additional analysis of model complexity and performance variations as dendritic branches increase, to further justify the rationality of dendritic branch selection.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:  \n1. Comprehensive theoretical foundation: The method analyzes RF neurons' limited frequency band response and the challenge of balancing energy efficiency with training efficiency.  \n2. Efficient Neural Dynamics: D-RF neurons achieve full-frequency coverage through multi-dendritic structures while ensuring parallel training and sparse spike generation.  \n3. Competitive Performance: The method performs well on datasets of varying lengths, demonstrating efficient temporal modeling capabilities.  \n4. Trade-offs between efficient and train efficiency: Ablation studies demonstrate D-RF's effective balance between energy efficiency and training efficiency.  \n\nWeakness:    \n1. Compared to conventional spiking neurons [1, 2], D-RF neurons exhibit complex neural dynamics that introduce additional computational overhead.  \n2. Although the paper validates the method on mainstream sequence tasks, it lacks performance evaluation on speech datasets, neuromorphic datasets, and other relevant domains.  \n\n[1] Spiking neuron models: Single neurons, populations, plasticity[M]. Cambridge university press.  \n[2] Accurate and efficient time-domain classification with adaptive spiking recurrent neural networks[J]. Nature Machine Intelligence.", "was_revised": true, "final_justification": "The rebuttal successfully addressed my concerns. I'll maintain my score and recommend this paper for acceptance.", "rebuttal": "Thank you very much for your valuable comments. We will further clarify and supplement the explanation to address your concerns.\n# W1: The Computational Overhead of D-RF Neuron.\nAnswer: As you per, D-RF neuron exhibits more complex dynamics, which significantly enhance the temporal processing capability of SNNs. On the S-CIFAR10 dataset, our method achieves an accuracy of **84.3%**, representing a **39.23%** improvement over the 45.07% achieved by the LIF neuron. Notably, current neuromorphic chips such as Loihi2 [1] supports complex-valued computations.\n\n# W2 & Q1: Evaluation on Speech and Neuromorphic Datasets.\nAnswer: To evaluate the generalizability of the proposed method, we further conducted experiments on the **Speech Dataset** (GSC) and the **neuromorphic dataset** (CIFAR10-DVS). The results are presented as follows:\n| Dataset | Method | Para | Acc(%)|\n|-|-|-|-|\n|GSC|LIF|106k|75.20|\n|GSC|**Ours**|123k|95.20|\n|CIFAR10DVS|PLIF|354k|74.80|\n|CIFAR10DVS|**Ours**|386k|86.20|\n\n# Q2: Algorithmic of D-RF Neuron.\nTo facilitate understanding and reproducibility, we provide both the **parallel training and serial inference** algorithms of the D-RF neuron.\n\n|Algorithm 1: Parallel Training of D-RF Neuron|\n|-|\n|1: **Input:** $\\mathcal{I}[t] \\in R$|\n|2: **Output:** $\\mathcal{S}[t] \\in R$|\n|3: $\\quad \\mathcal{Z}[t] = \\exp\\{\\delta \\mathcal{D}\\} \\mathcal{Z}[t-1] + \\Gamma^l \\mathcal{I}[t]$|\n|4: $\\quad V_{th}[t] = V_{pre} + \\sum_{k=1}^T \\alpha_k \\Theta \\left(\\Re\\{\\mathcal{Z}[t-k-1]\\} - V_{pre}\\right)$|\n|5: $\\quad \\mathcal{S}[t] = \\Theta(\\mathcal{C}\\Re\\{Z[t]\\} - V_{th}[t])$|\n\n|Algorithm 2: Serial Inference of D-RF Neuron|\n|-|\n|1: **Input:** $\\mathcal{I} \\in R^T, \\mathcal{K} \\in R^T$|\n|2: **Output:** $\\mathcal{S} \\in \\{0, 1\\}^T$|\n|3: $\\quad \\mathcal{Z} =\\mathcal{F}^{-1} \\{\\mathcal{F}\\{K\\} \\cdot \\mathcal{F} \\{I\\}\\} $|\n|4: $\\quad V_{th} = \\text{Conv1d}\\{\\Theta(\\mathcal{C}\\Re\\{\\mathcal{Z}\\} - V_{pre})\\}+V_{pre}$|\n|5: $\\quad \\mathcal{S} = \\Theta(\\mathcal{C}\\Re\\{Z[t]\\} - V_{th})$|\n\n# Q3: Analysis of Model Complexity and Performance.\nTo further justify the choice of dendritic branch number, we analyze both the **computational complexity** and **model size** of the D-RF neuron.\n**Computational Complexity**: Dynamics of the D-RF neuron is as follows:\n\n$$\\mathcal{Z}[t] = \\exp \\{\\delta \\mathcal{D}\\} Z[t-1] + \\Gamma^l \\mathcal{I}[t], \\quad \\mathcal{D} \\in \\mathbb{C}^{N\\times N}, \\Gamma^l \\in \\mathbb{R}^N,$$\n$$\\mathcal{S}[t] = \\Theta (\\mathcal{C} \\Re \\{ Z[t]\\} - \\sum_{k=1}^T \\alpha_k \\Theta \\left(\\Re\\{\\mathcal{Z}[t-k-1]\\} - V_{pre}) - V_{pre}\\right), \\quad \\mathcal{C} \\in \\mathbb{R}^N.$$\n\nTherefore, the computational complexity of the D-RF neuron is $\\mathcal{O}(NL)$, where $N$ denotes the number of dendrites and $L$ is the sequence length. The complexity increases linearly with the number of dendrites.\n\n**Model Size**: We evaluated the model’s accuracy on the S-CIFAR10 task under different numbers of dendrites.\n\n|Dataset|Metric|n=2|n=4|n=8|n=16|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|S-CIFAR10|Acc|82.1%|84.3%|84.6%|85.1%|\n||Para|209K|213K|216K|219K|222K|234K|\n\nAs shown in the table, the D-RF model achieves an accuracy of **84.3%** with 216K parameters when the number of dendrites is set to 4. Compared to $n=1$, this setting adds only 3K parameters while providing a 2.2% improvement in accuracy. In contrast, increasing to $n=8$ yields **only** a 0.3% gain in energy efficiency. Therefore, we choose $n=4$ as the optimal dendrite number for the S-CIFAR10 task, offering the **best trade-off** between model complexity and accuracy."}, {"review_id": "xVQpLjagzz", "reviewer": "Reviewer_kTeX", "summary": "The authors propose the Dendrite Resonate-and-Fire (DRF) neuron as a novel approach to address the limitations of conventional Resonate-and-Fire (RF) neurons. To overcome the limited bandwidth response of a single RF neuron, the DRF architecture integrates dendritic branches with varying eigen-frequencies to capture complex frequency compositions. Additionally, the model incorporates a dynamic adaptive threshold mechanism with parallelization to balance training efficiency and spike sparsity. Empirical evaluations demonstrate that the DRF neuron achieves consistently improved performance across multiple time-series datasets, while maintaining comparable model size, preserving spike sparsity, and enhancing training efficiency.", "questions": "1. How does the summation of dendritic signals at the soma contribute to frequency selectivity, given that the signals are merged before thresholding? How are multiple selected frequencies still reflected in the output spikes?\n\n2. What are the frequency initialization for the 8 dendrites that are shown in Figure 4 (c)? How is it possible to obtain enough information about the input signal with only 4-8 eigen-frequencies?\n\n3. What is the memory complexity of the D-RF network?", "limitations": "They have not addressed the memory usage required for parallel processing, which may be high since the whole sequence of the data is input simultaneously.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strength \n- Comparison of the time series dataset extensive and shows SOTA performance in all cases.\n- Faster training runtime and training energy cost efficient. Supports long sequence tasks. \n- Higher spike sparsity compared to other large spiking models on the LRA benchmark.\n\nWeakness\n- PathX is missing from the LRA Benchmark\n- The eigen-frequencies of the dendrites are fixed to the freq. bins of the input FFTs; in other words, depends on the discrete time delta. Would be interesting if these frequencies could be learned as well.\n- Neither hyperparameters nor the architecture of the models trained are mentioned in detail on the appendix or in the supplementary material; especially for S/PS-MNIST and SHD.   \n- (maybe too small a point to mention) Inconsistent number of parameters for S-CIFAR10 on Tables 7 and 8.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable suggestion. To address your concern, we have conducted additional experiments.\n# W1: PathX is missing from the LRA Benchmark\n\nSome existing SNN-based sequence models, such as Spiking LMUFormer [1] and PRF [2], encounter training difficulties or fail to converge on the Path-X task. To ensure a fair comparison, we **only** report results on the commonly tasks. Nevertheless, to address your concern, we conduct additional evaluations on the Path-X task:\n\n|Method|Acc(%)|\n|:-:|:-:|\n|Spiking LMUFormer [1]|Fail|\n|PRF [2]|Fail|\n|Binary S4D [3]|61.2%|\n|Ours|**82.8%**| \n\n[1] Lmuformer: Low complexity yet powerful spiking model with legendre memory units[C]. ICLR 2024.\n\n[2] PRF: Parallel Resonate and Fire Neuron for Long Sequence Learning in Spiking Neural Networks.\n\n[3] Learning long sequences in spiking neural networks[J]. Scientific Reports, 2024.\n\n# W2: The eigen-frequencies of the dendrites are fixed to the freq. bins of the input FFTs; in other words, depends on the discrete time delta. Would be interesting if these frequencies could be learned as well.\n\nIt is important to clarify that both the frequency bands and discrete time parameters of the D-RF neuron are **learnable**. Specifically, for the D-RF neuron:\n\n$$\\mathcal{Z}[t] = \\exp \\{\\delta \\mathcal{D}\\} Z[t-1] + \\Gamma^l \\mathcal{I}[t], \\quad \\mathcal{D} \\in \\mathbb{C}^{N\\times N}, \\Gamma^l \\in \\mathbb{R}^N,$$ \n\n$$\\mathcal{S}[t] = \\Theta (\\mathcal{C} \\Re \\{ Z[t]\\} - \\sum_{k=1}^T \\alpha_k \\Theta \\left(\\Re\\{\\mathcal{Z}[t-k-1]\\} - V_{pre}) - V_{pre}\\right), \\quad \\mathcal{C} \\in \\mathbb{R}^N,$$\nwhere $\\delta$ and $\\mathcal{D}$ are **learnable** parameters, enabling the D-RF neuron to more effectively capture information from the input signal.\n\n# W3: Neither hyperparameters nor the architecture of the models trained are mentioned in detail on the appendix or in the supplementary material; especially for S/PS-MNIST and SHD.\n\nWe adopt a residual block-based network architecture, where each block consists of: \"D-RF Module – 1×1 Convolution – Spiking Neuron – 1×1 Convolution.\" The detailed configurations for each task are as follows:\n\n|Tasks|S/PS-MNIST|SHD|S-CIFAR10|ListOps|Text|Retrieval|Image|Pathfinder|PathX|\n|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:|\n|Depth|3|3|8|6|6|6|6|6|6|\n|Channel|128|128|256|128|256|256|512|256|256|\n|Para|155.1K|155.1K|216K|297K|841K|1.1M|3.2M|1.3M|846K|\n|Acc|99.50 / 98.20|96.20|84.30|60.02|86.52|90.02|85.32|92.36|82.8%|\n\n# W4: (maybe too small a point to mention) Inconsistent number of parameters for S-CIFAR10 on Tables 7 and 8.\nThank you for your correction. After recalculation, the parameter count for the S-CIFAR10 task is **216K**. We will further revise the manuscript accordingly.\n\n# Q1: How does the summation of dendritic signals at the soma contribute to frequency selectivity, given that the signals are merged before thresholding? How are multiple selected frequencies still reflected in the output spikes?\n\nFor the D-RF neuron, its dynamics can be expressed as follows:\n\n$$\\mathcal{Z}[t] = \\exp \\{\\delta \\mathcal{D}\\} Z[t-1] + \\Gamma^l \\mathcal{I}[t], \\quad \\mathcal{D} \\in \\mathbb{C}^{N\\times N}, \\Gamma^l \\in \\mathbb{R}^N,$$\n$$\\mathcal{S}[t] = \\Theta (\\mathcal{C} \\Re \\{ Z[t]\\} - \\sum_{k=1}^T \\alpha_k \\Theta \\left(\\Re\\{\\mathcal{Z}[t-k-1]\\} - V_{pre}) - V_{pre}\\right), \\quad \\mathcal{C} \\in \\mathbb{R}^N.$$\nThe learnable dendritic parameters $\\mathcal{D}$ and $\\delta$ determine the neuron's response to specific frequency bands, enabling **frequency-selective filtering**. The parameter $\\mathcal{C}$ adjusts the relative weighting of each frequency band. The soma integrates the weighted dendritic inputs and generates sparse spikes through an adaptive threshold mechanism. During training, all parameters are optimized via gradient-based methods to achieve task-specific frequency selection.\n\n# Question 2: What are the frequency initialization for the 8 dendrites that are shown in Figure 4 (c)? How is it possible to obtain enough information about the input signal with only 4-8 eigen-frequencies?\n\nThe eight dendrites are initialized using a linear distribution. For the $k$-th input, the initial frequency is defined as follows:\n$$\\omega_k = \\frac{k}{8}\\pi, k \\in \\[0,1,2,\\cdots, k-1\\],$$\nThrough the **learnable** frequency parameters, the model adaptively adjusts to an optimal frequency-band distribution. As shown in Fig. 4c, the trained dendrites nearly cover the entire frequency spectrum, ensuring comprehensive extraction of frequency-domain information. The performance analysis for different numbers of dendrites is presented below:\n\n|Dataset|Metric|n=1|n=4|n=8|n=16|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|S-CIFAR10|Acc|80.3%|84.3%|84.6%|85.1%|\n|S-CIFAR10|Para|209k|216k|222k|234k|\n|ListOps|Acc|55.2%|59.1%|60.2%|60.3%|\n|ListOps|Para|275k|280k|288k|310K|\n\nThe results show that when $n=4$ or $n=8$, the D-RF model achieves an effective balance between energy efficiency and parameter count.\n\n# Question 3: What is the memory complexity of the D-RF network?\nTo further illustrate the memory requirements of the D-RF model, we provide an analysis of its memory usage and computational complexity:\n\n|Model|Period|Memory Complexity|Time Complexity|\n|:-:|:-:|:-:|:-:|\n|LIF|Train|O(1)|O(L^2)|\n|LIF|Inference|O(1)|O(L)|\n|D-RF (Ours) |Train|O(NL)|O(LlogL)|\n|D-RF (Ours) |Inference|O(N)|O(L)|\n\nHere, $N$ denotes the number of dendrites and $L$ the sequence length. As shown in the table, during training, the D-RF model leverages parallel computation to accelerate long-sequence processing. During inference, its space complexity is only $\\mathcal{O}(N)$, introducing minimal additional memory overhead compared to LIF neurons."}, {"review_id": "PXlwaxbtTQ", "reviewer": "Reviewer_m2wP", "summary": "This paper proposes a D-RF neuron to address efficiency and performance challenges in long sequence modeling. The model comprises multi-dendritic branches and a soma structure, where each dendritic branch leverages RF neuron oscillatory properties to capture frequency-specific signals, while the soma employs an adaptive threshold mechanism that dynamically adjusts thresholds based on historical spiking activity. Experimental results demonstrate that D-RF achieves competitive performance across multiple long sequence tasks while maintaining sparse firing and training efficiency.", "questions": "1. Does introducing a multi-dendritic structure into LIF neurons enhance their temporal modeling capabilities? The authors are required to conduct experiments to verify this.\n2. D-RF neurons seem to introduce complex number operations. Will this affect their deployment in neuromorphic chips?\n3. How to train the RF neuron-based neural networks? The authors should clarify this question.", "limitations": "Yes.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strengths:\n1. The paper provides a systematic analysis of RF neuron limitations, identifying constrained frequency response bandwidth and energy-efficiency trade-offs as key bottlenecks.\n2. The multi-dendritic architecture effectively expands frequency response capabilities, while the adaptive threshold mechanism facilitates efficient optimization.\n3. The adaptive threshold design maintains O(L log L) computational complexity, enabling practical SNN training for extended sequence modeling.\n4. Extensive experiments thoroughly validate the proposed method's effectiveness and efficiency.\n\nWeaknesses:\n1. Insufficient analysis of computational overhead analysis comparing D-RF to conventional spiking neurons.\n2. D-RF introduces additional complex number operations, and the authors did not discuss whether this feature hinders the deployment of neuromorphic chips.\n3. The selection of dendritic branch number lacks systematic analysis of the performance-complexity trade-off.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the valuable comments. In response to the concerns raised, we provide the following detailed replies and additional experiments:\n# Weakness 1: Insufficient analysis of computational overhead analysis comparing D-RF to conventional spiking neurons.\nWe compared the computational complexity of the D-RF neuron and the LIF neuron.\n\n**Dynamics of the LIF Neuron**: \n\n$H[t] = \\tau U[t-1] + \\frac{1}{\\tau} (I[t] - (U[t-1] - V_reset)),$\n\n$S[t] = \\Theta(U[t] - V_{th}),$\n\n$U[t] = H[t] (1 - S[t]) + V_reset S[t],$\n\nwhere $I[t]$ denotes the input current, $V_{\\text{reset}}[t]$ represents the reset membrane potential, and $\\tau$ is the time constant. For a sequence of length $T$, the computational complexity of the LIF neuron is $\\mathcal{O}(T)$.\n\n**Dynamics of the D-RF Neuron**: \n\n$\\mathcal{Z}[t] = \\exp \\{\\delta \\mathcal{D}\\} Z[t-1] + \\Gamma^l \\mathcal{I}[t], \\quad \\mathcal{D} \\in \\mathbb{C}^{N\\times N}, \\Gamma^l \\in \\mathbb{R},^N$  \n\n$\\mathcal{S}[t] = \\Theta (\\mathcal{C} \\Re \\{ Z[t]\\} - \\sum_{k=1}^T \\alpha_k \\Theta \\left(\\Re\\{\\mathcal{Z}[t-k-1]\\} - V_{pre}) - V_{pre}\\right), \\quad \\mathcal{C} \\in \\mathbb{R}^N,$\n\n$N$ denotes the number of dendritic branches. Therefore, the computational complexity of the D-RF neuron is $\\mathcal{O}(NT)$.\n\n#  Weakness 2 and Question 2: The impact of D-RF's complex number operations on neuromorphic chip deployment remains inadequately discussed by the authors.\n\nAs you per, D-RF model introduces additional complex-valued computations. It is worth noting that modern neuromorphic chips already support such operations, such as Loihi2 [1]. This provides hardware-level support for the deployment of the D-RF model.\n\n[1] Efficient neuromorphic signal processing with loihi 2[C]. SiPS 2021.\n\n# Weakness 3: The selection of dendritic branch number lacks systematic analysis of the performance-complexity trade-off.\n\nTo further address your concern, we conducted ablation experiment on the S-CIFAR10 dataset:\n\n|Dataset|Metric|n=2|n=4|n=8|n=16|\n|:-:|:-:|:-:|:-:|:-:|:-:|\n|S-CIFAR10|Acc|82.1%|84.3%|84.6%|85.1%|\n|S-CIFAR10|Para|209K|213K|216K|219K|222K|234K|\n\nThe model reaches 84.3% accuracy at $n=4$, with only 3K more parameters than $n=2$ and a 2.2% accuracy gain. Increasing to $n=8$ brings just a 0.3% improvement over $n=4$. Thus, $n=4$ offers the best trade-off between performance and complexity on the S-CIFAR10 tasks.\n\n# Question 1: Does introducing a multi-dendritic structure into LIF neurons enhance their temporal modeling capabilities? The authors are required to conduct experiments to verify this.\nAs you noted, multi-dendrite modeling significantly enhances the temporal processing capability of neurons. To assess this effect, we replace the D-RF neuron with the LIF neuron and evaluate performance on the S-CIFAR10 task:\n\n|Method|Para|Acc|\n|:-:|:-:|:-:|\n|LIF|204k|45.07|\n|LIF with Dendritc|215k|68.10|\n|D-RF (Ours)|216k|84.30|\n\nExperimental results show that incorporating a multi-dendrite structure significantly improves the modeling capability of the LIF neuron. Based on this, the D-RF neuron achieves an additional **16.2%** performance gain, further highlighting the advantage of RF neurons in temporal processing tasks.\n\n# Question 3: How to train the RF neuron-based neural networks? The authors should clarify this question.\n\nIn this work, we adapt the STBP training method [2] to effectively address the issue of non-differentiable gradients. The weight gradient is defined as follows:\n$$\\frac{\\partial L}{\\partial W } = \\sum_t \\frac{\\partial L}{\\partial S[t]} \\frac{\\partial S[t]}{\\partial V[t]} \\frac{\\partial V[t]}{\\partial W},$$\n$\\frac{\\partial S[t]}{\\partial V[t]}$ denotes the surrogate gradient of the Heaviside function. In this work, we use the arctangent function [3] as the surrogate gradient. \n$$\\frac{\\partial S[t]}{\\partial V[t]} = \\frac{\\alpha}{2(1+(\\frac{\\pi}{2}\\alpha x)^2))}.$$\n\n[2] Spatio-temporal backpropagation for training high-performance spiking neural networks[J]. Frontiers in neuroscience, 2018.\n\n[3] Temporal efficient training of spiking neural network via gradient re-weighting[C]. ICLR, 2022."}, {"review_id": "mbwCmQHNuQ", "reviewer": "Reviewer_zzoM", "summary": "The authors describe the resonate and fire (RF) neuron in the context of being able to model longer term dependencies than the more conventional LIF neuron.  They propose a dendritic RF (D-RF) neuron where some of the dynamics are modelled by dendrites, allowing a single neuron to handle multiple resonances (one per dendrite).  They further propose an adaptive threshold that can be thought of as analogous to the spike frequency adaptation that is common in LIF implementations.\nExperiments on basic but reasonable datasets yield good results.", "questions": "One assumes (perhaps wrongly) that the original RF neuron was based on observation of biology.  To what extent do the proposed modifications retain (or not) that biological plausibility?\n\nHow does the sequence length compare with current long NLP baselines?  Speech input (at ~100 Hz sample rate) or repeated sampling of static examples surely represent long inputs, but how long is long?", "limitations": "No, I don't see limitations addressed explicitly in the paper.  However, the work is speculative enough to not really merit such an analysis.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\nPaper structure is good; subject is timely and well motivated.\nThe RF architecture is relatively not well studied, so the general approach is quite novel.\nResults are good.\n\nWeaknesses:\nThe modifications to RF are in the spirit of functions that are known to work well on LIF neurons, so the novelty could be dismissed as obvious.\nThe datasets used are not especially \"long\" when compared to the things that are used for LLMs nowadays.  Then again, the LLM state of the art is well advanced whereas this is much more speculative.\nIt is not obvious whether the proposed architecture remains biologically plausible.", "was_revised": true, "final_justification": "I think this is basically a good paper. The authors responded well to allay most of my concerns so I've bumped up the recommendation to accept.", "rebuttal": "# Weakness 1: The modifications to RF are in the spirit of functions that are known to work well on LIF neurons, so the novelty could be dismissed as obvious. \n\nThe proposed adaptive threshold mechanism fundamentally differs from spike frequency adaptation [1–2] in both implementation and computational complexity, despite their conceptual similarity.\n\n**Implementation-wise**: The adaptive threshold mechanism employs causal convolution to efficiently integrate historical spikes during inference, whereas spike frequency adaptation relies on a sequential adjustment strategy, which limits its ability to parallelize.\n\n**Computational Complexity**: The adaptive threshold mechanism achieves $\\mathcal{O}(L \\log L)$ complexity during training, significantly more efficient than the $\\mathcal{O}(L^2)$ complexity of spike frequency adaptation.\n\nExperimental results demonstrate that for sequences of length 4096, the adaptive threshold mechanism yields a **148×** improvement in training efficiency over traditional approaches.\n\n[1] Spike frequency adaptation: bridging neural models and neuromorphic applications[J]. Communications Engineering, 2024.\n\n[2] Advancing spatio-temporal processing through adaptation in spiking neural networks[J]. Nature Communications, 2025\n\n# Weakness 2: The datasets used are not especially \"long\" when compared to the things that are used for LLMs nowadays. The LLM state of the art is well advanced whereas this is much more speculative. \n\nAs you per, current ANN-based LLMs are capable of processing sequences ranging from 16K to 1M tokens [3–4]. However, such performance gains largely rely on **complex architectural designs** and **large parameter scales**, leading to significant computational overhead. We solve the problem from the perspective of neural dynamics, with the goal of enhancing memory and temporal processing at the neuron level. This enhancement is intended to enable a better trade-off between energy efficiency and performance, particularly under smaller model scales.\n\nD-RF model provides a **viable component** for building low-power SNN-based LLMs. Future work will further investigate its modeling capability on large-scale language tasks.\n\n[3] GPT-4 Technical Report. OpenAI, 2023.\n\n[4] Qwen2.5 Technical Report. Qwen, 2024.\n\n# Weakness 3 & Question 1: One assumes (perhaps wrongly) that the original RF neuron was based on observation of biology. To what extent do the proposed modifications retain (or not) that biological plausibility? \n\nThe proposed D-RF neuron is inspired by two key neurobiological observations: the **Multi-dendritic Structure** of cortical neurons and their inherent **Resonance Characteristics**. \n\n**Multi-dendritic Structure:** Neuroscientific research shown that cortical neurons are primarily composed of dendritic cells, accounting for approximately 80% of all neurons, and their complex dendritic arborizations play a crucial role in information processing [5–6]. \n\n**Resonance Characteristics:** Biological studies demonstrate that resonance is an intrinsic property of neurons, characterized by subthreshold membrane potential oscillations that enable selective responses to inputs at specific frequencies [7–8]. \n\n**Design of D-RF Neuron**: The D-RF neuron serves as a computational abstraction of these two biological mechanisms, aiming to maintain biological plausibility while enhancing computational efficiency. Each dendrite is modeled as a damped oscillatory dynamic, enabling efficient extraction of frequency-specific features. Additionally, the multi-dendrite and soma architecture mimics the parallel processing ability of biological neurons, allowing the model to capture multi-band information from input signals. This design achieves a favorable trade-off between biological fidelity and computational simplicity, supporting scalable deployment in large-scale neural networks.\n\n[5] Pyramidal cell types drive functionally distinct cortical activity patterns during decision-making[J]. Nature neuroscience, 2023.\n\n[6] Brain-wide presynaptic networks of functionally distinct cortical neurons[J]. Nature, 2025.\n\n[7] Oscillations emerging from noise-driven steady state in networks with electrical synapses and subthreshold resonance[J]. Nature communications, 2014\n\n[8] Resonance with subthreshold oscillatory drive organizes activity and optimizes learning in neural networks[J]. PNAS, 2023.\n\n# Question 2: How does the sequence length compare with current long NLP baselines? Speech input (at ~100 Hz sample rate) or repeated sampling of static examples surely represent long inputs, but how long is long?\n\n**Compare with long NLP baseline**: We surveyed commonly used text benchmarks, including GLUE, WikiText-103, and WikiText-2. A summary is provided in the table below:\n\n|Benchmark|Tokens|\n|:-:|:-:|\n|GLUE|10~60|\n|Wikitext103|1024~2048|\n|Wikitext2|512~1024|\n|Ours|1024~4096|\n\nTherefore, the sequence length used in this work is comparable to that of existing NLP benchmarks, indicating its potential for effective language modeling.\n\n**how long is long**: Due to the inherent training challenges of SNNs—such as gradient approximation and the instability introduced by BPTT—most existing models adopt **4** to **8** timesteps to maintain a balance between performance and training stability. Within this constraint, tasks like S/PS-MNIST (784 steps) and S-CIFAR10 (1024 steps) are commonly regarded as long-sequence benchmarks. In this context, the proposed D-RF model further demonstrates its capability to model extended sequences, achieving effective performance on the **LRA** benchmark with 1K–4K time steps.\n\n# Paper Formatting Concerns\nThank you for the correction. Reference [23] will be revised accordingly.\n\n[23] Higuchi S, Kairat S, Bohté S M, et al. Balanced resonate-and-fire neurons[C]//Proceedings of the 41st International Conference on Machine Learning. 2024: 18305-18323."}]

Author rebuttal:

Meta-review: During the rebuttal, the authors addressed many of the reviewers' concerns and the reviewers have reached a collective positive grade. I am hence recommending an acceptance.
Decision: Accept (poster)