Beyond Linear Processing: Dendritic Bilinear Integration in Spiking Neural Networks
Authors: Jingyang Ma
Chongming Liu
Songting Li
Douglas Zhou
Venue: ICLR 2026 Poster (2026)
Keywords: spiking neuron models
spiking neural networks
dendritic integration
brain-inspired computing
Primary area: foundation or frontier models, including LLMs

Abstract: As widely used neuron model in Spiking Neural Networks (SNNs), the Leaky Integrate-and-Fire (LIF) model assumes the linear summation of injected currents. However, recent studies have revealed that a biological neuron can integrate inputs nonlinearly and perform computations such as XOR while an LIF neuron cannot. To bridge this gap, we propose the Dendritic LIF (DLIF) model, which incorporates a bilinear dendritic integration rule derived from neurophysiological experiments. At the single-neuron level, we theoretically demonstrate that a DLIF neuron can capture input correlations, enabling it to perform nonlinear classification tasks. At the network level, we prove that DLIF neurons can preserve and propagate correlation structures from the input layer to the readout layer. These theoretical findings are further confirmed by our numerical experiments. Extensive experiments across diverse architectures—including ResNet, VGG, and Transformer—demonstrate that DLIF achieves state-of-the-art performance on static (CIFAR-10/100, ImageNet) and neuromorphic (DVS-Gesture, DVS-CIFAR10) benchmarks, surpassing LIF and other advanced alternatives while maintaining comparable computational cost. This work provides a biologically plausible and computationally powerful spiking neuron model, paving the way for next-generation brain-inspired computing.

Published as a conference paper at ICLR 2026
BEYOND LINEAR PROCESSING: DENDRITIC BILINEAR INTEGRATION IN SPIKING NEURAL NETWORKS
Jingyang Ma Chongming Li Songting Lishe", and Douglas Zhou
" "School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China IIsttitute of Natural Sciences, Shanghai Jiao Tong University, Shanghai, China *Ministry of Education Key Laboratory of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai, China a SSangghai Frontier Science Center of Modern Analysis, Shanghai Jiao Tong University, Shanghai,
China
SState Key Laboratory of Synergistic Chem- Bio Synthesis, Shanghai Jiao Tong University, Shanghai, China
ABSTRACT
As widely used neuron model in Spiking Neural Networks (SNNs), the Leaky Integrate-and-Fiin (LIF) model assumes the linear summation of injected currents. However, recent studies have revealed that biological neuron can integrate in- puts nonlinearly and perform computations such as XOR while an LIF neuron cannot. To bridge this gap, we propose the Dendritic LIF (DLIF) model, which incorporates a bilinear dendritic integration rule derived from neurophysiologi- cal experiments. At the single-neuron level, we theoretically demonstrate that a DLIF neuron can capture input correlations, enabling it to perform nonlinear classification tasks. At the network level, we prove that DLIF neurons can pre- serve and propagate correlation structures from the input layer to the readout layer. These theoretical findings are further confirmed by our numerical exper- iments. Extensive experiments across diverse VGG, and mance static (CIFAR-10/100, ImageNet) and neuromorphic (DVS-Gesture, DVS-CIFAR1O) benchmarks, surpassing LIF and other advanced alternatives while maintaining comparable computational cost. This work provides biologically plausible and computationally powerful spiking neuron model, paving the way for next-generation brain-inspired computing. All source code available com
ResNet, that DLIF achieves state-of-the- perfor-
https: httt:s///gtaaa....ooa/eeeaaoaggg
INTRODUCTION
Spiking neural networks (SNNs) increasingly recognized the next generation of neural network paradigm that closely emulates biological neural systems through discrete spike- based communication between neurons (Maass 1997b. Unlike traditional artificial neural networks (ANNs), which operate on continuous-valuee activations (Deng et al. 2020). SNNs employ event-driven computation via discrete spikes. This fundamental difference enables SNNs to achieve significantly more efficient and sparse data processing, offering substantial energy efficiency ad advantages over conventional ANNs (Roy et al. 2019 Davies al. 2018 Pei al. 2019 Ma al. 2022).
Most SNNs employ the Leaky Integrate-and-Firg (LIF) neuron model, a simplified abstraction of biological neurons that omits dendritic processing. In biological neurons, dendrites receive and integrate multiple input currents before transmitting them to the soma. The LIF model assumes the linear summation of the input currents. However, numerous studies have revealed that the integration process on the dendrites is nonlinear (Polsky et al., 2004 Poirazi et al. 2003 Ujfalussy et al. 2018 Beniaguev et al. 202Tp, which plays a critical role complex computations of biological
Corresponding author, songting @ @aj.ingssssuuu.u...n 7d7 sjtu.edu.cn

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neurons, such direction selectivity (Branco et al.
2010),
coincidence detection (Agmon-Sin
et al. 1998), and logical operations (Gidon et al.
2020). Consequently. incorporating nonlinear dendritic integration features into spiking neuron models is increasingly important direction in
brain-inspired computing (Pagkalos et al. 2024 Acharya et al. 2022
In this paper, propose novel spiking neuron model, termed the Dendritic Leaky Integrate- and-Fire (DLIF) model, which is based on a bilinear dendritic integration rule observed in recent experiments (Hao et al. 2009 Li et al., 2014 2019). Theoretically, we show that the single-neuron level a DLIF neuron can capture input correlations, while at the network level DLIF neurons preserve and propagate correlation structures across layers, with these results further validated by numerical experiments. We also show that, across various tasks and deep neural network architectures, using DLIF models can significantly improve the performance of SNNs compared to those with LIF models and other spiking neuron models, with no significant increase in computational cost.
The main contributions of this paper are summarized as follows:
1. We propose the Dendritic Leaky Integrate-and -nd-Fire (DLIF) model, a biologically plausible spiking neuron model that incorporates a bilinear dendritic integration rule observed in neurophysiological experiments.
2. We theoretically establish, and numerically confirm, that DLIF neurons can capture input correlations to perform nonlinear classification at the single-cell level, and preserve and propagate these correlations through the network.
3. We demonstrate that, across multiple architectures, DLIF neurons achieve an average accu- racy of 88%, with a 1.23% improvement over conventional LIF-based SNNs (83.95%), and state-of-the-as performance on both static and dynamic vision benchmarks. This improvement is obtained with only 0.17 7 mJ energy overhead (a 3.05% relative increase).
4. We show that the DLIF models possess computational advantages comparable to those of other advanced spiking neuron models, including PLIF, GLIF, EIF, QIF and DH-LIF.
RELATED WORK
Bilinear Neural Networks. Several studies have explored bilinear neural networks in conventional
ANNs. A line of work has focused on feature fusion and pooling for visual recognition (Lin et al.
2013 Gao et al. 2016 Kong & Fowlkes) 2017). Another direction has investigated bilinear neurons as arcchtecttrrll primitives: one-rank bilinear neurons (Yun et al.. 2019), pixel-wise bilinear filters (Zoumpourlis al.) 2017). and bilinear networks with stabilized training strategies (Fan al.
2025a). In addition, bilinear formulations have also been applied in other domains, including MRT reconstruction (Ahmed al., 2022) and low-rank structures (Pearce al. 2025). (Qi & Wang 2022] further highlighted that bilinear networks can achieve substantially higher efficacy and efficiency than conventional neural networks. Our work differs by introducing bilinear dendritic integration into the spiking neural network framework and providing theoretical analysis of its role in preserving input correlations, perspective absent prior bilinear ANN models.
Models and Algorithms Inspired by Dendritic Computation. Recent studies have increasingly incorporated dendritic computation principles into machine learning frameworks. Some works have drawn inspiration from dendritic cable theory, local learning rules, and dendritic event-based processing (Bicknell & Häusser 2021 Payeur et al.] 2022 Sacramento et al.) 2018) Yang et al., 2021). Others have focused network structures inspired by dendritic compartmentalization and connectivity (Guerguiev et al. 2017; Chavlis & Poirazi 2025 Gao et al.) 2018). Meanwhile, practical applications of dendritic integration have been demonstrated in diverse architectures, such as convolutional networks with dendritic modules (Liu et al.) (2024) and dendritic artificial neural networks (Egrioglu & Bas, 2024). Together, these works highlight the growing importance of dendritic principles powerful source of inspiration for advancing machine learning.
Neuron Models in SNNs. Various extensions of the standard LIF model have been proposed
to enhance the representational capacity of SNNs. Some works introduce additional flexibility in neuronal dynamics, such as learnable time constants or adaptive thresholds (Fang et al. [201]] Bellec
et al. 2020 Feng al. 2022 Chen et al. 2022). Others enrich the computational structure of
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spiking neurons through gating mechanisms, soft reset strategies, or membrane potential rectifiers (ao et al. 2022: Guo et al. 2022b). Multi-compartmemt and multi-branee models further capture dendritic or temporal heterogeneity (Zheng al. 2024 Wang al. 2025a Liu et al., (222b). while multi-synaptic formulations enable simultaneous integration at different scales (Fan et al. (025b). These diverse extensions underline the central role of neuron model design in advancing the power of SNNs. However, to the best of our knowledge, no prior work has sought to optimize spiking neuron models through the bilinear form of dendritic integration.
3 DENDRITIC LEAKY INTEGRATE-AND-FAE (DLIF) MODEL
Biological neurons have complex dendritic structures that are responsible for receiving multiple external inputs, integrating them, and transmitting the processed signals to the soma (Stuart et al. 2016). The spiking neuron model faithfully replicates key functional properties of biological neurons through temporal integration of input signals and subsequent generation of output spikes (Gerstner et al. 2014). The sub-threshold somatic membrane potential (t) of spiking neuron is always governed by:
dV - - Vrest) + RI((),
dt
where μ represents the time constant, Viest is the resting potential, R denotes the resistance, and T(t) is the input current. When ((t) reaches a certain firing threshold ViA. the neuron emits a spike and resets the potential back to Vreat- The resulting output spike train Y(t) is formally expressed as (t) >; 5(t t¹), where ö is the Dirac delta function, and marks the i-th firing time of the neuron. As the commonly used model in SNNs, the LIF model assumes linear summation of external inputs (Burkitt 2006), i.e.
100) = Σ w;s w s(t..
((t)
where (w,) denotes the synaptic weight from the pre-synaptic neurons to the target post- synaptic neuron, and st) = (t) represents the {0, 1] spike trains from pre-synaptic neuron. However, biological experiments indicate that dendrites integrate inputs in nonlinear manner (Polsky ct al. 2004; Spruston 2008). This dendritic nonlinearity is essential for various computational functions, such as direction selectivity (Branco et al., 2010), coincidence detection (Agmon-Sun et al. 1998), and logical operations (Gidon et al. 2020). Consequently, the linear dendritic integration mechanism of LIF models fails to fully capture the complex characteristics of biological neurons and cannot perform the rich nonlinear computations. To address this limitation, we propose a novel spiking neuron model, termed the Dendritic Leaky antegrate-and-Fire (DLIF) model.
3.1 FORMULATION OF THE DLIF MODEL.
Recent neurophysiological experiments and theoretical analysis have demonstrated that the dendritic integration of synaptic inputs by single neuron is not linear, but conforms to bilinear form (Hao et al. 2009 Li et al. 2014, 2019). This bilinear integration property can be characterized by considering two synaptic imputs and b, where the dendritic integration yields not just the linear sum a + b but includes an additional bilinear interaction term kab. Thus, the total integrated input becomes b+ kab. Here, & is referred is independent of the input intensities and only dependent on the relative spatial positions of the two synaptic inputs. Consequently, when neuron receives multiple synaptic input spike trains s(t) = (t) with connection weights w, there will be additional bilinear integration terms (t)ss, (t) (1 <issije S n). associated with a symmetric bilinear coefficient matrix K (Ki)), whose diagonal entries are zero. The integrated input can then be expressed as:
the bilinear dendritie integration coefficient, which
1(t) Σ (t)
+
Σ 1= 1>
M 2K (t)s, = w s(t) + (t))((tt.
Note that (t)s, (t) can be directly realized through an AND operation, Eq. 3 won't introduce any additional multiplication operations. This preserves SNNs' computational efficiency, as their
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baske-baeed communication naturally favors additive operations multiplicative (Roy al. 2019). Then the dynamics of the somatic membrane potential in the DLIF model can be described as:
(t) = (V((1) - VVres) + Rllw" s(t) s" (())))))))
dt
3.2 THEORETICAL ANALYSIS OF DLIF'S COMPUTATIONAL ADVANTAGES
first demonstrate the advantage of the DLIF neuron model from a theoretical perspective. We consider a binary classification problem where each input sample is represented as a binary matrix, where each column corresponds to the spike trains N input neurons at given time step: = s()),, s(r)] E {0, 1) xxx,,,, where N N and N 2, and is the total time steps. Two input classes with distributions Dj and D2 have identical mean firing rates but distinct pairwise correlations:
Es [1,] = -_s_ =c
Es-D, [SS = Ci + -B
where 1, is the all-ones column vector with length We claim that single DLIF neuron discriminate between two input classes by exhibiting distinct firing rates in response to them. (Gerstner & Kistler 2002)) have demonstrated that for spiking neuron models with dynamics of the form given in Eq. in the input regime where the current is sufficient to elicit spiking, the output firing rate is proportional the average input current. Therefore, in this regime, significant difference in firing rates is equivalent to a difference in input currents. Suppose the time-averaged input current to DLIF neuron is denoted I, we can obtain the following theorem (See proof in Section A.I
Theorem 1. Let two input spike train distributions D1 and 'D2 be defined as in Eq. 5 Then there always exists bilinear coefficient matrix K which can distinguish two corresponding input currents to the DLIF neuron, i.e.,
= [B[riDi] B[1[D2]| >
Moreover, under the constraint
|||r S 1, the optimal choice of K that maximizes 5T is given by
C1 C2
K = + | C2
This theorem shows that DLIF neurons can exploit input correlation structures for classification via the bilinear matrix K. However, for multi-layer SNNs, it is generally reasonable to assume that correlated inputs appear at the input layer. In what follows, further show that a two-layer SNN with DLIF ncurons can preserve input correlations in the hidden layer.
Building upon the same input spike train distributions as in Eq. (5), without loss of generality,
assume that |[ - C22 || 1. These spike trains are encoded by MM hidden neurons, and we denote the output spike trains of the hidden neurons Y [1(1), (2). ... y(T) f0, 10 MxT, For each class, we define the correlation matrix P. of Y as P. = ES YY for C (1, and denote the resulting matrices for LIF and DLIF neurons by pl.r PL PDLIF PPDLI, respectively, Let W [w.,..., W W A be the weight matrix and K K ......, Kar with K,, NxN be the bilinear coefficient matrices. We suppose that ||w;|| < and K S for 1, 1.... M. Then we obtain the following theorem (See proof in Section A.I).
Theorem 2.
Let the input spike trains be drawn from distributions D2 and D2 defined in Eq. (5) Then for any choice of synaptic weight matrices LIF and WDLFF there exists bilinear coefficient matrices K for the DLIF neurons such that
|PDLIF
PDLIF
2 O.
and furthemmoee,
IIPLIF
PDLIF
||
|PLLIP - PLIF
This theorem demonstrates that DLIF networks are more capable than LIF networks preserving the correlation structures inherent in the input data. Combined with Theorem [ which proves that the DLIF neuron model can classify inputs with correlated structures, these theoretical derivations collectively guarantee the superior computational and representational power of DLIF neurons over LIF neurons. Next, further validate the above theorems through several numerical experiments.
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A
B
E
XOR operation
pathway 2
on
psthwwy 2
pathway1
pathway2
off
pethway 1
off on pathway 1
C
LF
9.8
Irpact hoae
Irput intor
9.0
0.4
0.4
02
02
200
400
600
800
1000
4.5
6
30
0.0
5.
4
9
0.0
timesteps
Figure I: A single DLIF model can effectively solve the XOR problem.
(A): A neuron receives inputs from distinct pathways (represented in blue and red), each containing five synaptic inputs.
(B): The XOR operation schematic: the neuron remains inactive when both pathways are on or off, but it fires when only one pathway is on, thereby implementing the XOR operation. It is trivial that the neuron is inactive when both pathways are off. Therefore, we only consider the other three input patterns, which are labeled as !. @ and ④ respectively.
(C): Expectation of the normalized difference in correlation matrices between two classes
Cr-Ce CC-Call
(D): Expectation of the bilinear coefficient K for the DLIF neuron.
(E): Three different patterns the XOR problem. Pathway includes synapse indices 1-5, while pathway 2 includes synapse indices 6-10. The DLIF model keeps silent when both pathways are activated (labeled as ①) and fires when receiving input patterns with only one pathway is activated (labeled as ② and )), thus achieving the XOR operation (orange). In contrast, the LIF model still fires when both pathways are active (labeled as !), thus it unable to perform the XOR operation.
3.3 NUMERICAL VERIFICATION
Verification of Theorem 1 We first verify Theorem I by showing that single DLIF neuron can implement the XOR operation, consistent with recent biological findings that individual neurons can perform such computations (Gidon et al. 2020), whereas standard LIF neuron cannot (Mostafa 2017). In our simulation, the neuron receives inputs from two pathways (five synapses each) with identical Poisson firing rates (Fig. A)). The target is to fire when exactly one pathway is active but remain silent otherwise (Fig. TB). We group the three non- trivial input patterns into two classes: Class 1 (both pathways active) and Class (only one pathway active). Under this setup, the spike trains of the two input classes follow a similar distribution as in Eq. (5). We train w for LIF and both w and K for DLIF to minimize the mean squared error between actual and target firing rates. As shown in Fig. E the DLIF model successfully performs XOR while the LIF model fails, and the learned bilinear matrices K closely match C.-Ca
121FF (Fig. C. D), consistent with theoretical predictions.
Verification of Theorem 2 To validate Theorem we design controlled numerical experiments both low and moderate input dimensionalities. First, two-dimensional case provides a simple and interpretable setting, where two Poisson input spike trains are received by two hidden neurons which employs either LIF or DLIF neurons. The network trained maximize output correlation (details in the Appendix), and we compute the normalized correlation between output spike trains. As shown in Fig. 2A,, DLIF neurons preserve input correlations substantially more effectively than LIF neurons. To further examine whether this advantage persists in more complex scenarios, consider a een-dimensional case where two distinct input classes with distributions D2 and D2 as in Eq. (5.. These inputs fed into dindeennsional hidden layer, and the training procedure is similar to the 2D case. As illustrated in Fig. [B, DLIF neurons again yield significantly larger separation between the output correlation matrices of the two classes compared LIF neurons. Together, these
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A
Comparisen
1P
Frobenlus norms across
1.0
0.261
0.038x
8.000
UF DUIF
y
0.8
Outpue Splie Correlaioo
0.2-
Frobenius norm
0.6
0.6-
0.4
0.2
0.0
0.2
0.6
1.0
0.0
5
0
200
400
600
800
10000
2 Density
Input Spike Correlation
Density
Simulation trial
Figure 2: DLIF neurons preserve and propagate input correlations.
(A): For the two-dimensional case, we compare the input-output correlation across 1000 simulation trials for both LIF and DLIF neurons. Each dot represents individual trial. The regression slopes indicate that DLIF neurons preserve input correlations more effectively (slope 0. 0929, R= 0.494) than LIF neurons (slope = 0.038, R=0.261).
(B): For the ten-dimensional case, we compare the Frobenius norm of the difference between the output correlation matrices of two input classes across 1,000 simulation trials for both LIF and DLIF neurons. Each dot corresponds to an individual trial. The scatter plot, together with the marginal distributions, demonstrates that DLIF neurons consistently produce a larger separation between input classes than LIF neurons.
results confirm the theoretical prediction in Theorem 2 that DLIF neurons more effectively amplify and propagate correlation differences across layers.
4 PERFORMANCE OF DLIF MODELS IN SNNS
We further investigate whether the advantage of DLIF models can be scaled to large-scale SNN architectures. The detailed SNN architectures and experimental setup used in this work are presented in the Appendix.
As shown Eq. the DLIF model introduces additional bilinear parameters K. Through both biological recordings and computational modeling, Li et al. (2019) reported that dendritic bilinear interactions are inherently sparse (== 90%). Motivated by this, we adopt sparsification scheme in which only small fraction of coefficients are trainable. Specifically, choice that both biologically inspired and empirically validated by an ablation study as shown in Section 4.4 In the following experiments, compare not only model accuracy but also parameter count, FLOPs, and energy consumption (detailed calculation methods are provided the Appendix).
set sparsity level to 90%,
4.1 EXPERIMENTS IN LARGE-SCALL SNNs FOR STATIC DATASETS
We first evaluate the performance of the DLIF models in the SNNs on static image classification benchmarks, including CIFARIO (Krizhevsky et al. 2009). CIFAR 100 (Krizhessky et al.) 2009) and ImageNet (Deng et al. 2009) datasets. Additionally, we test the DLIF models' applicability across diverse network architectures and training methods. Specifically, we explore the integration of DLIF models in large-scale SNNs, including VGG (Simonyan Zisserman 2014), ResNet (He et al.) 2016) and Transformer (Ashish al. 2017). We also experiment using different learning paradigms including SLTT (Meng et al. 2023), ESG (Guo et al. 2022a). OTTT (Xioo et al., 2022), STBP-tdBN (Zheng al. 2021). TET (Deng et al. 2022), ESL (Liu al. 2025a). TSER (Yu et al. 2025a), FSTA Yu et al. (0225)) SSSA (Wang et al.) 2025b), Spike-driven Transformer Yao et a.. 2023a) and Fota-Spi Yao al. 2024), as proposed in previous works.

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Table 1: Results Static Datasets
Params
FLOPs
Energy Cost
Mean Std (%)
Dataset
Method
Network
Neuron
(M)
(G)
(mJ)
LIF
12.08
1.82
1.77
94.44.10.21
SLTT
ResNet-18
DLIF
12.20
1.83
1.78
95.51:10.47
LIF
12.61
1.94
1.88
93.16
STBP-tdBN
ResNet-19
DLIF
12.78
1.95
1.89
94.35:10.27
LIF
12.08
1.83
1.77
96.39
ESL
ResNet-18
DLIF
12.20
1.83
1.78
96.72-10.11
LIF
138.08
15,48
15.05
95.01=+0.10
CIFARIO
TSER
VGG-16
DLIF
136.02
15.94
15.57
95.97
LIF
12.61
1.94
1.88
96.52:10.09
FSTA
ResNet-19
DLIF
12.78
1.95
1.89
96.91=00008
LIF
5.57
1.21
1.04
96.10
SSSA
Transformer
DLIF
5.68
1.28
1.10
96.81=1.117
Spike-driven
LIF
9.32
1.08
1.05
95.60
Transformer
Transformer
DLIF
10.05
1.10
1.07
96.22-10.17
LIF
12.17
1.82
1.77
74.38+10.30
SLTT
ResNet-18
DLIF
13.21
1.83
1.78
76.89-10.29
LIF
138.44
15.50
15.07
70.18:00.09
ESG
VGG-16
DLIF
146.86
16.04
15.60
1..52.10.16
LIF
123.60
7.63
7.42
71.05:10.04
OTTT
VGG-11
DLIF
136.71
8.13
7.91
74.71
LIF
138.44
15.50
15.07
77.06=10.04
CIFAR100
TSER
VGG-16
DLIF
146.86
16.04
15.60
78.37+00.14
LIF
12.74
1.94
1.88
80.42+00.09
FSTA
ResNet-19
DLIF
12.91
1.95
1.89
80.97-0.12
LIF
5.57
1.21
1.04
80.10
SSSA
Transformer
DLIF
5.63
1.30
1.12
80.58:10.09
Spike-driven Transformer
LIF
9.35
1.08
1.05
78.40
Transformer
DLIF
10.12
1.10
1.07
79.46:10.32
LIF
21.80
3.66
3.56
64.79
TET
ResNet-34
DLIF
23.42
3,85
3.74
67.32:10.39
LIF
21.80
3.66
3.56
63.72
STBP-1dBN
ResNet-34
DLIF
23.42
3.85
3.74
66.79-10.53
LIF
138.44
15,50
15.00
74.32
ESL
VGG-16
DLIF
146.86
16.04
15.60
75.11-10.28
LIF
21.80
3.66
3.56
03.16:10.15
ImageNet
TSER
ResNet-34
DLIF
23.42
3.85
3.74
73.82-10.29
LIF
21.80
3.66
3.56
70.23+0.12
FSTA
ResNet-34
DLIF
23.42
3.85
3.74
71.06
LIF
53.7
36.75
35.75
80.23
SSSA
Transformer
DLIF
57.37
37.04
36.11
80.75
LIF
55.40
53.92
52.40
80.00
Meta-SpikeFormer
Transformer
DLIF
58.73
54.35
53.68
80.57+10.28
Bold values represent the best results for each method
As summarized Table [ our results reveal consistent and substantial accuracy improvements attributable to DLIF models across all configurations. Specifically, DLIF-based SNNs achieve absolute accuracy gains of 0.33%-1.19% on CIFAR-10, with even more pronounced improvements of 3.66% CIFAR- 100 and 4.52%-3.0%% ImageNet. Notably, these performance enhancements come with only minimal energy overhead, quantified as just 0.23 mJ average increase (2.61% % relative to LIF implementations).
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Table 2: Results Neuromorphie Datasets
Params
FLOPs
Energy Cost
Mean Std
Dataset
Method
Network
Neuron
(M)
(G)
(mJ)
(%)
LIF
123.24
7.61
7.40
98.50:10.21
SLTT
VGG-11
DLIF
125.96
7.85
7.63
98.92-+0.24
LIF
123.24
7.61
7.40
96.88
OTTT
VGG-11
DLIF
125.96
7.85
7.63
97.43
LIF
11.74
1.71
1.67
96.87
DVS-Gesture
STBP-tdBN
ResNet-17
DLIF
11.87
1.72
1.68
98.05::0.41
LIF
27.48
2.13
2.08
94.91
SSNN
VGG-9
DLIF
28.55
2.23
2.18
96.27
Spike-driven
LIF
2.59
0.36
0.35
99.30
Transformer
Transformer
DLIF
3.02
0.37
0.36
99,43 0.27
LIF
123.24
7.61
7.40
82.20-
SLTT
VGG-11
DLIF
125.96
7.85
7.63
83.74+0.62
LIF
12.61
1.94
1.88
67.8
STBP-tdBN
ResNet-19
DLIF
12.78
1.95
1.89
70.88:10.45
LIF
27.48
2.13
2.08
78.57
SSNN
VGG-9
DLIF
28.55
2.23
2.18
80.85::0.42
DVS-CIFARIO
LIF
13.57
2.21
2.17
82.70= 10
FSTA
ResNet-20
DLIF
13.85
2.38
2.31
82.98-00113
LIF
1.52
0.54
0.52
82.30
SSSA
Transformer
DLIF
1.84
0.56
0.53
82.91
Spike-driven
LIF
2.59
0.36
0.35
80.00
Transformer
Transformer
DLIF
3.02
0.37
0.36
81.76:10.27
Bold values represent the best results for each method
4.2 EXPERIMENTS LARGE- SNNs NEUROMORPHIC DATASETS
In contrast to static image datasets, neuromorphic datasets like DVS-Gesture (Amir et al. 2017 and CIS-CIFARIO (Li et al. 2017) naturally encode temporal information, thereby better show- casing SNNs' inherent advantages processing spatiotemporal patterns. We further evaluate the performance of SNNs using DLIF models versus LIF models across different network architectures including including VGG ResNet and Transformer, and different learning paradigms such SLTT, OTTT, STBP-dBNN SSNN, Spike-driven Transformer and FSTA.
The experimental results in Table demonstrate consistent performance gains when using DLIF models. Specifically, on DVS-Gesture, DLIF-based SNNs achieve 0. 4.13%-1.333 1.36% higher accuracy, while on the more complex DVS-CIFARIO, the improvements reach 0.28%-3.08%% Importantly, these significant accuracy gains come with only 0.10 mJ average energy increase (3.24% relative to LIF models), further validating DLIF's practical utility in neuromorphic computing applications.
In addition, we further compare the training time and memory cost of DLIF and LIF models in Section A.3 The results show that DLIF increases per-epoch training time and GPU memory usage by roughly 10%, but this overhead remains acceptable for large-scale SNN training.
Beyond static and neuromorphic benchmarks, we further evaluate DLIF models in reinforcement learning (RI. by integrating them into deep spiking Q-network (DSQN) (Chen et al.. (024). On Atari games, the based DSQN outperforms its LIF counterpart, demonstrating the flexibility of DLIF models to adapt effectively across diverse task paradigms (see Section A.4 for details).
4.3 COMPARISONS WITH OTHER SPIKING NEURON MODELS
Several studies have proposed modifications to the existing LIF models in SNNs. For instance, Fang et al. (2021) introduced the Parametric Leaky Integrate-and -nn--Fere (PLIF) model, which included learnable time constants to enhance heterogeneity. YYao et al. (2022) proposed the Gated Leaky Imeerrate-andFiin (GLIF) model, incorporating gating units into LIF models to improve their rep-
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Table 3: Comparisons with Other Point Spiking Neuron Models.
Accuracy CIFAR-10(%)
Accuracy
Accuracy ImageNet(()) %)
Accuracy DVS-CIFARIO($) DVS-Gesture(
Accuracy
Neuron Model
10FAR-100(%%
PLIF
93.50
69.26
74.80
97.92
GLIF
95.03=00.08
77.35:10.07
69.09
78.10
QIF
92.98=0.14
75.91=0.00
67.49-10.25
73.27=:0119
96.18=0.11
EIF
93.08-0.117
67.1410.30 78.27+00.39
76.18:1..15
76.271:0.39
97.01=0.18
DLIF 95.78=0.21
71.27=:0.24
80.46+1.17
98.61
Bold values represent the best results for each dataset; indicates result is not reported
Table 4: Comparisons with DH-LIF.
Neuron Model
Accuracy SHD(%)
Accuracy on SSC(%)
DH-LIF
92.10
82.46
DLIF
92.71
83.13
Bold values represent the best results for each dataset
resentation capacity. addition the PLIF and GLIF models, other variants introduce nonlinear operations to the LIF model, such as the Quadratic Integrate-and-Fing (QIF) model and the Exponen- tial Integrate-and-Fing (EIF) model (Gerstner al. 2014). The detailed of the dynamics of the QIF and EILF model are shown in the Appendix. To ensure fair comparison, we adopt the same network architectures and hyperparameter configurations as in prior works. Across CIFAR-10, CIFAR- ImageNet, DVS-CIFARE and DVS-Gesture, our results (Table 3) show that DLIF consistently outperforms these advanced point-neuron models.
In addition to point-neuron models, we also compare DLIF with multi--ompprtment spiking neuron model-the DH-LIF proposed by (Zheng et al. 2024). which incorporates temporal dendritic heterogeneity. We follow the experimental setup in (Zheng et al. 2024) and conduct comparisons on two spiking speech recognition datasets SHD and SSC (Cramer et al. 2020). Under comparable parameter settings, DLIF consistently surpasses DH-LIF both tasks, demonstrating its effectiveness relative to dendritic neuron models as well (Table 4
4.4 ABLATION STUDY
Sparsity Level We conducted a systematic ablation study by varying sparsity levels from 0% to 100% the CIFAR- 100 dataset with the ResNet- 18 architecture trained using SLTT. As summarized in Table 5 the ACC/FLOPE ratio consistently reaches its maximum at 90%. This indicates that 90% sparsity provides the most favorable trade-off between computational efficiency and predictive performance. Combined with biological evidence that dendritic bilinear interactions are naturally sparse about 90% (Li et al. 2019), these results justify our adoption of 90% sparsity.
Table 5: Ablation Study of Sparsity Level
Sparsity level(%)
0
15 30
45
60
FLOPs(()
1.92
1.905
1.89
1.875
1.86
ACC(%)
78.67
78.45
78.14
77.42
77.26
(CC/)
40.97
41.18
41.34
41.29
41.54
Sparsity (evel(%)
75
85
90
95
100
FLOPs(G)
1.845
1.835
1.83
1.825
1.82
ACC(%)
76.33
76.38
76.89
74.61
74.38
(CC/)
41.67
41.62
42.02
40.88
40.87
Bold values represent the best results

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Published as a conference paper at ICLR 2026
Bilinear Coefficients To further illustrate the role of the bilinear coefficients K in the DLIF model, we conduct ablation studies to assess their impact. As shown Section A.3 removing the bilinear coefficients- either before or after training consistently reduces test aceuracy, confirming their critical importance in the DLIF model. In addition, our structured- and experiments show that low-rank formulations, while offering stronger compression, lead clearly weaker performance, whereas the diagonal-block and random sparse formulations perform compara- bly under matched sparsity levels. Together with sparsity-level ablations showing that performance peaks around 90% sparsity, these findings reinforce the central role of the bilinear coefficients K in the DLIF model and highlight that 90% random sparse matrix provides a simple yet effective parameterization.
5 DISCUSSION AND CONCLUSION
This paper proposed the DLIF model, which incorporates a biologically inspired dendritic bilinear integration rule into spiking neurons. While (Li er. 2019) investigated bilinear dendritic integration using conductance-base model with voltage-dependent synaptic dynamics, this formulation is difficult to scale to large SNNs. contrast, our DLIF model adopts current-based abstraction that removes these biophysical dependencies and enables efficient, scalable implementation while preserving the bilinear rule, We further provided theoretical guarantees and numerical verification that DLIF neurons can exploit input correlations for nonlinear computation and preserve correlation structures across layers. Experiments on static, neuromorphic and RL benchmarks consistently showed that DLIF can achieve superior performance over LIF and other advanced spiking models, with minimal additional energy cost.
There remain some important avenues for further research. One direction is extend DLIF beyond vision and RL tasks to natural language processing, especially in light of the rapid progress of large language models. Another direction concerns hardware adaptation. While DLIF models demonstrate strong algorithmic efficiency, adopting them onto neuromorphic hardware will be crucial to fully exploit the low-power and low-latency advantages of SNNs. Addressing these open challenges would further enhance the applicability and impact of DLIF models.
In summary, we propose novel spiking neuron model for SNNs that enhances computational capabilities at both the single-neuron and network levels. We believe this work provides solid foundation for the design and application of future brain-inspired computing.
REPRODUCIBILITY STATEMENT
We ensure reproducibility at several levels. First, the DLIF model clearly described in the main text, including its mathematical formulation and theoretical analysis. Second, the assumptions and complete proofs all theorems are provided in Appendix. Third, experimental settings-including datasets, architectures, hyperparameters, and training paradigms are specified in Appendix.
7 THE USE OF LARGE LANGUAGE MODELS (LLMs)
We used L.MSs to polish the manuscript, for example by improving phrasing and checking spelling and grammar, L.MSs were also employed to assist in literature search and discovery, such as by providing keywords to retrieve related works. However, the core ideas, methodology. and contributions of this paper were conceived independently and did not rely on LLMs.
ACKNOWLEDGMENT
This work was supported by STI2030-Major Projects 2021ZD0200204 (D.Z., S.L.): Shanghai Munic- ipal Commission of Science and Technology with Grant No. 24JS28 10400 (D.Z. S.L.); National Natural Science Foundation China Grant 12271361, 72250710674 (S.L.); National Natural Science Foundation of China with Grant No. 92570202, 12225109 (D.Z.), Lingang Laboratory Grant No. LG- AG-QS202202-01 (S.L., D.Z.); and the Student Innovation Center at Shanghai Jiao Tong University (J.M., C.I.,, S.L.,, D.Z.).
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Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 6023-6032, 2019.
Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guoqi Li. Going deeper with directly-trained larger spiking neural networks. Proceedings of the AAAI conference artificial intelligence, 35, 2021.
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A APPENDIX
A.1 PROOFS OF THEOREMS
We consider a binary classification problem where each input sample is binary matrix: S ((2), s(r) E {0, 10,1) where N E N and N 2 2, and 7 is the total number of discrete time steps. Two input classes with distributions D1 and D2 have identical mean firing rates but distinct pairwise correlations:
Es [S1]] -E =
(A1)
Eg-D, [SS = C1 + ESWD, [551
where 1, is the all-ones column vector with length (Gerstner & Kistler 2002) have demonstrated that for spiking neuron models with dynamics of the form given in 7 dV(()
(V(t) ( Vrest) (t),,,
in the input regime where the current is sufficient to elicit spiking, the firing rate is proportional to the average input current. Assuming that the average firing rate is sufficiently high induce spiking, we obtain the following theorem
Theorem A1, Let two input spike train distributions D1 and D22 be defined as in Eq. 5 Then there always exists bilinear coefficient matrix K which distinguish two corresponding input currents to the DLIF neuron, i.e.,
81 [B[TiPi] B[T[D2]| >
Moreover, under the constraint
| < 1, the optimal choice of K that maximizes 51 is given by
K = C - C2
|| - C2
Proof. Let and K denote the weight vector and the bilinear coefficient matrix of DLIF neuron. respectively. With certain input S, the total integrated input current is defined as in Eq. 3 Then the average input current is:
1= Σ 1(1)= - - E=
(w"SS, S1, + tr KS))
(A2)
Then the difference of the input current between the two input classes is:
8T [[] B D2] = Ir (K(( -C))
(A3)
Since C1 # C22 there exists at two entries 1 -2) (C1 C2) # 0 symmetric). Consider a bilinear coefficient matrix K that places nonzero values only on these two entry (e.g., Kij K j; 1 and all other entries zero). Then ò7 2k 0, which establishes the existence of such that 5T > O.
C2 is
Furthermore, under the normalization constraint | F
have
[Ir(K -C2))
K F C C2llr
|C1 -C2l
(A4)
Equality holds if and only if
C₁ - C2
K
C C211
in which case 87 attains its maximal value || - C2ll F. Therefore, only is separation guaranteed, but there also exists an optimal K that maximizes the expected difference in input current, ensuring the clearest distinction between the two classes.
At the network level, suppose the two input classes still characterized by Eq. A1 These inputs are encoded by M hidden neurons, and we denote the input current to the hidden neurons IR Mxr, Assume that the two classes exhibit significantly different For each class, we
as I [i()), (.,)),,..,,(.)) correlation structures; without loss of generality, we let |[ C2ll define the correlation matrix of Q. of I as Q. LESDD [TT for (1,2), and denote the
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resulting matrices for LIF and DLIF neurons by Q - Q Q respectively. Let W [W',.... Warl E R MxN be the weight matrix and K = K K M with Km E RNN be the bilinear coefficient matrices. Without loss of generality, assume that the LIF and DLIF neurons share the same synaptie weight matrix W = [w....., WM) E RMxN Suppose further that the weight vectors and bilinear coefficient matrices normalized such that ||w:]|r and | S for all = M. Then we obtain the following lemma:
Lemma A1. Let the input spike trains be drawn from distributions DD and D2 defined in Eq. (5) Then for any choice of synaptic weight matrices W, there exists bilinear coefficient matrices K for the DLIF neurons such that
QIQ Q2
|1 1=2C
20> O.
and
1QPII - 12LTII1 > |IQ1
Q |r-
Proof.
The input current ith hidden neurons are defined
I = w S,
PLLP = w 2,(),
(A5)
where (s(S)) [s(1)] K (() ..... S(T) Kis(T)). For class 2). define the correlation matrix as
(Q%) = ES-D. (E,)))
(A6)
where {LIF, DLIF).
Step 1 (LIF case)
For LIF neurons we have
(Qu )if
E.
1 (1) (T)))))= ESSD. [ww SS (sss,,]] [n,]= = C.w,
(A7)
where C. = R..DD. [SS*] is the input correlation matrix of class Hence
LLFF bj
(Q)
Q Jy w (Ci W2)
(A8)
Step 2 (DLIF case)
For DLIF neurons, we have
(Q DLIF Jij =E D. 1;
(Q
DLIF (I (LIF)T )T
(A9)
EESDD. D. (ww SS wy + (,(S)) wy w w (S) (())
We abbreviate
Giy(c) Es (5)s)) wj + w S (S)",,
Hcy(c) = EgnD. (S)))
Then
DLIF A
= (QPLIF
QDLIF) )ij = ES-D m SS w;] Es-D, SS" wy]
LIF
(A10)
Giy(1) (2)
Hij -1,(2)
Hincar-bllinaa term
billcar-bilincar term
Since (t) E 10, we have st(1) s:(t) for any N We can define (t())s((t) as the second-orden moments for + j. (t)sx() as the third-order moments for + j * le and so
For the linear- bilinear term, we have
Gij(c) = - Σ K (0))))))))))
Σ
K ())))))))))))))))) (A11)
1=1 p:4
pzg
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Since the higher-order moments negligible compared to second order moments, then we have
Giz(c) Σ pl((K)) + (K; pm Wjj + (Kj (K))pywp Wip + (Kj) Win) sp(()s((1)] (A12)
PFQ
=1 -1,,[( + lw @ Kj + (w,1" 1" + +ww3) 3 K, C C.}1
where is the Hadamard product. And we have
A = G (1) - City = 1n [1[[(w;1" [[f[w,1++10] +w Ko + o , (A13)
Similarly, for the biliear-bilineen term, we have
Hi)(c) = ΣΣ (K)
(A14)
1=1 potq ustN
Since the higher-order moments are negligible compared to second-order moments, then we have
Hijlc) (K.)) (K) pm (p())s((t)
(A15)
= =1 (K₁ K, 1 C.)1
4 H,y(1 H,j(2) 12 の K2 o (Ci C2)
(A16)
Step 3 (Norm comparison)
Since
A DLIP A #4 + +
(A17)
WLOG, we suppose that ALF 2 0. Since ||C C2 = 1, there exists p # q such that k
(Ci C2)m O. We select K; Kj alepe + e then
A 2a²k
(A18)
A = (Wip + Wig + wip + wig )ak
Since a = is one of the zeros of A + AG there exists a small e such that A + A > when and satisfying
|| |
K 1. Therefore, we have
|A DLIF
LIF | ij
+ A G ~
(A19)
ij
and
DLIF
(A20)
A
DLIF
LIF + A 1 >0
(A21)
1/
When ANF < 0, we can select K; a(epe" + e and K, = ea + t,,e)) then
20 k
(A22)
A (-Wip win + Wip win)ak
Similarly, since 0 is still one of the zeros of A# + AG there exists small such that A# A when and satisfying
| K F < 1. Therefore, have
|APLP
LIL
+ A C + > C >
(A23)
and
DLIF
A
LIF
(A24)
DLIF 1
A
LIF 6
A 1j
A >
(A25)
In general, we have
||
DLIF
2 C > 0.
(A26)
and
A
DLIF
|| > |
LIF
(A27)
This completes the proof.
n
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The above lemma demonstrates that, compared to LIF neurons, DLIF neurons better preserve the correlation of the original input spike trains at the current-ipput level. (De La Rocha et al., 2007) > further proved that, for spiking neurons of the form given in Eq. T the correlation of the output spike train is positively correlated with that of the input current, leading to the following lemma.
Lemma A2. Consider two spiking neurons defined by Eq. Q Let the correlation of their input currents be in the expectation sense. Then, there exists constant k > 0 such that the correlation of their outuu spike trains, also measured in expectation, satisfies b = ka.
The proof of Lemma A2 can refer to (De La Rocha al. 2007).
Then we denote the output spike trains of the hidden neurons as Y = (1)(1, (2)) ... ,((T) e (0. 1) MXT_ For each class, we define the correlation matrix P. of Y as P. ESSDD (Y') for {1,2}, and denote the resulting matrices for LIF and DLIF neurons by
PLAF PLIF.. PDLIP, PDLIF, respectively. Let W [ wM) E RMXX be the weight matrix and K [K...... KM] M] with Km E NxN be the bilinear coefficient matrices. We suppose that ||w,ll F S and || I for i = =. M.. Then obtain the following theorem
Theorem A2. Let the input spike trains be drawn from distributions D1 and D2 defined in Eg. (Al
Then, for any choice of synaptic weight matrices WIF and WLIF, there exists bilinear coefficient matrices K for the DLIF neurons such that
-P
DLIF
2C>00 0.
and
ILPLIF_
P2
PLIFFIIE
IF
|
P.I.FIle.
Proof.
According to Lemma Al and Lemma A2
we have
PLIF
DDLIF
|| k: Q DLIF Q
2 20>0 O.
(A28)
IPPLIF -POLIF
||
k
QPII QD
(A29)
klIQPIF 11F 1
IP-.I-
-P..F
A.2
DETAILS ABOUT SPIKING NEURON MODELS AND NETWORKS
A.2.1 DYNAMICS
The sub-threshold somatic membrane potential (t) is governed by:
dV dt - (V((() Vrest) + RI((),
(A30)
For computational implementation, we set R and the time interval to 1, yielding the discrete dynamics equations:
U 4) 1]+ +
S([] H[ ViA),
(A31)
V = =(1 -1 S[()UA + Sees
where U[[ and [[[] represents the membrane potential before and after reset operations, respectively. 1 1...3..... - denotes the time step, and is the time duration. H(x) is the Heaviside step function. In our experiments, Ven and Vrest are set to be I and 0, respectively. The choice of is specified in the supplementary.
Eq. (31 defines the update rule for neuronal dynamics in SNNs. Notably, SNN architectures maintain structural parallels with conventional AÑNs, permitting direct adaptation of established frameworks such VGG, ResNet and Transformer through substitution of activation functions with spiking neuron models. Standard operations including convolution and pooling remain fully compatible.
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Table 6: Comparison of Training Time and Memory Cost
Neuron Model
Network Architecture
Training Time
Memory Cost
Accuracy
LIF
ResNet-34
1.54h
20.17G
70.23%
LIF
ResNet-50
1.8lh
23.54G
70.79%
DLIF
ResNet-34
1.66h
22.39G
71.06%
SNNs support both neuromorphic and static image data processing. While neuromorphic data naturally contains temporal information, static images require conversion temporal inputs via pixel value encoding (Rueckauer et al., (211)) Diehl & Cook, 2015) Shrestha & Orchard 2018). The network output is determined by the highest average firing rate among output layer neurons, which corresponds to the predicted class label.
A.2.2 TRAINING ALGORITIIM
Training spiking neural networks faces a fundamental challenge: the non-differentiabiiity of spike sequences, which prevents direct gradient-base based optimization. This limitation is overcome by com- bining Backpropagation Through Time (BPTT) with surrogate gradient methods (Huh & Sejnowski 2018 Shrestha & Orchard 2018 Wu et al. 2018 2019). The -indeefferee term RV can be approximated by the surrogate functions such as rectangle or triangle functions (Wu et al., 2018): 0s
BV =sgn((V - Veal < $ or ΦV oz max(0, VV VeA I), where a hyperparameter and sgn is the sign function. With the surrogate gradient, the gradient-based algorithm can be applied to train SNNs.
A.3 EXPERIMENTS OF COMPARING TRAINING TIME AND MEMORY COST
We conducted additional measurements comparing the training time and memory overhead of DLIF with the standard LIF-based SNN. To ensure a fair evaluation, we performed experiments on ImageNet using the ResNet-34 architecture trained with the standard BPTT algorithm, with a batch size of 256, running on a single NVIDIA A100 GPU. The reported training time corresponds to the wall-clock time per epoch. The wall-clock training time and memory cost are computed based on the first three epochs of training.
The results in Table show that DLIF increases per-epoch training time by 8% and GPU memory consumption by 11%. However, this overhead remains acceptable for large-scale SNN training. To further demonstrate that the performance improvement of DLIF is not merely due an increase in parameter count, we also conducted experiments using the LIF model with ResNet-50. In this setting, the training time and memory consumption are both higher than those the DLIF-based ResNet-34, yet the accuracy remains lower than that achieved by DLIF-ResNet-34.
A.4 EXPERIMENTS IN LARGE-SCALE SNNS FOR REINFORCEMENT LEARNING TASKS
Reinforcement learning represents a fundamental machine learning paradigm where agents learn optimal decision-maie making policies through environmental interactions, guided by reward signals (Kaeibling et al.) 1996: Wiering & Van Otterlo 2012). The Atari game has emerged as a standard benchmark for evaluating RL algorithms. challenging agents maximize game scores through pixel- level inputs (Mnih et al. 2015). While Q-learning Waakkins & Dayan 1992) and its deep learning variants have demonstrated strong performance, recent work has successfully adapted spiking neural networks for Q-value approximation, enabling efficient processing of high-dimensional state spaces (Patel et al. 2019 Tan al. 2021 Chen et al. 2024).
In our experiments, we implement DLIF models within a deep spiking Q-network (DSQN) architec- ture and evaluate the performance across 16 Atari games (Chen et al. 2024). A comparative analysis with the LIF-based DSQN reveals that DLIF models achieve an average performance improvement of 20.62% in final game scores across all tested environments as shown in Table 7 Due to slight variations in imput and output dimensions across different games, the computational cost exhibits minor fluctuations. On average, the DSQN model based on the LIF neuron has 0.21M parameters, 0.83M FLOPs, and an energy cost of p.81 In comparison, the Daae-based DSQN incurs 0.23M
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Table 7: Results Atari Games
Game
DSQN with LIF (Mean Std)
DSQN with DLIF (Mean Std)
Atlantis
2515926 + 73782.9
2942971.5
Beam Rider
5327.6 6 + 178.0
7306.9 544.6
Boxing
82.7 + 8.7
10.5
Breakout
368.1 + 9.8
47..1
Crazy Climber
95164.4 + 1232.9
100673.6-1519.2
Gopher
4233.1 + 176.5
6151.41219.4
Jamesbond
469.4 = 82.7
587.2=84.7
Kangaroo
5824.4 = 540.8
7374.2-702.9
Krull
6991.1 + 107.0
7883.5
Name this game
6981.0 = 192.8
804122
Pong
19.5 + 0.4
20.3 0.6
Road Runner
27725.6 + 3954.0
29401.1 +5007.1-507..1
Space Invaders
1209.9 + 61.2
2769.6 6 105.3
Star Gunner
1657.8 8 + 102.0
1984.4 136.1
Tutankham
266.5 + 12.2
294.2 11.5
Video Pinball
408032.6 + 41687.8
436980.0 41047.2
Bold values represent the best results for each Atari game
Table 8: Ablation Study of Bilinear Coefficients
Dataset
Method
Network
Neuron
Mean Std(%)
LIF
64.79
TET
ResNet-34
DLIF
67.32:10.39
DLIF*
25.83:22.67
LIF
63.72
ImageNet
STBP-tdBN
ResNet-34
DLIF
66.79-10.53
DLIF
21.36 1.89
LIF
80.00
Meta-SpikeFormerr
Transformer
DLIF
80.57 0.28
DLIF#
17.12+5.81
Bold values represent the best results for each method;* means removing K after training
parameters (+0.002M, 9.52%), 0.9M FLOPs (+0.07M, +8.43%), and an energy cost 0.86pJJ (40.05p.1, These modest increases in computational and energy costs justified by the corresponding performance gains.
A.5 ABLATION STUDY
To highlight the contribution of the bilinear coefficients K in the DLIF model, we perform ablation studies on the ImageNet dataset under three training paradigms: TET, STBP-1dBN, and Meta- SpikeFormer. We consider two ways of removing the bilinear matrices: (i) removing K before training, which reduces the DLIF model standard LIF model, and (ii) removing K after training to assess their impact on the learned representations. As shown in Table both settings consistently lead drop test accuracy across all paradigms, confirming the critical role of bilinear coefficients in the DLIF model.
To further examine how different parameterizations of the bilinear matrix K influence DLIF per- formance, we evaluated two structured variants-a diagonal-block sparse matrix and a low-rank factorization, motivated by biological or computational considerations. In addition to the random 90% sparse matrix used in our main experiments, constructed (i) diagonal-bloc' matrix with bandwidth n 8, matched to the same overall sparsity level 90%)) and (ii) a low-rank factorization 0), which removes K
K = UVT with ranks 1, 2, We also include the zero-rank
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Table 9: Comparison of different parameterizations of the bilinear matrix
Matrix Type
ZR
LR (r=1)
LR (r=2)
IR (r=3)
DB
RS
#Params (M)
12.17
12.30
12.43
12.56
13.21
13.21
ACC (%)
74.38
74.55
74.82
75.27
76.82
76.89
ZR=zero rank; LR=low rank; DB=diagonal block; RS-random sparse
entirely and reduces DLIF standard LIF model. All variants were evaluated CIFAR-100 using ResNet- 18 trained with SLTT.
The results in Tiihhght two distinct conclusions. First, low-rank parameterization substantially reduces the number of bilinear parameters but yields noticeably weaker performance, with accuracy consistently below the sparse formulations. Second, the diagonal-bloee matrix performs comparably to the random 90% sparse matrix, suggesting that incorporating locality structure neither improves nor degrades performance under similar sparsity levels. These observations indicate that while our current sparse formulation provides effective balance between accuracy and parameter efficiency, exploring richer biologically inspired structural priors for K remains an interesting direction for future investigation.
A.6 IMPLEMENTATION DETAILS
A.6.1 PARAMETER COUNT, FLOPS, ENERGY COST
In addition to reporting accuracy, we also evaluate models using three standard metries: parameter count, FLOPs, and theoretical energy consumption. The parameter count is obtained by summing all trainable weights in the network. When computing the parameter count, we explicitly account for the sparsity of the bilinear matrix-only the non-zero entries after masking are included as trainable parameters. To enforce this sparsity practice, we generate fixed binary mask M before training, where 90% of the entries are set to zero and the remaining 10% to one. During training, the bilinear matrix is parameterized as K- = @ M where @ denotes the Hadamard product. This procedure ensures that the effective bilinear parameters strictly follow the desired sparsity pattern, and only the unmasked entries contribute to the parameter count and optimization. For the ResNet and VGG architectures, we incorporate the bilinear operation into the convolutional layers of the first two blocks. For the Transformer architecture, we apply the bilinear operation to the FFN layers in the first encoder blocks.
FLOPs are estimated by counting all multiplication and addition operations required in a single forward pass across all layers. Theoretical energy consumption is then computed as the weighted sum of these operations, where each multiplication incurs an energy cost of EMAC and each addition
incurs an energy cost of EAC. Following prior work(Kundu et al. 2021 Lemaire et al. 2022 Yao
et al. 2025: Zhou et al., 2023 Yao et al. 2023b Hu et al. [2024), we adopt EMAC = 4.6 p. and EAC 0.9 pl under 45nm CMOS eechnolgyyHorowiin 2014). Accordingly, the total theoretical
(EMAC NMAC + EAC NNc), where NMAC and NAC denote the total number of multiplications and additions during one forward pass, T the number of timesteps, and is the average spiking firing rate.
energy of an SNN is given by EsNN
A.6.2 OTHER SPIKING NEURON MODELS
The dynamics the PIIF, GLIF and DH-LIF models can be found in (Fang et al. 2021), Yao et al. 2022) and 211eppzhn respectively. The dynamics of the QIF model defined
du = co(v (v vee) + RT
(A32)
dt
while the dynamics of the EIF model are given by:
du
1 - Wit
-v Urest) + Ar exp
+ RI
(A33)
dt
AT
The QIF and EIF models are configured with the following hyperparameters: Uyest = 0, 0, = 0.8. an 1, and AT = 1. For fair comparison, evaluate the performance of QIF, EIF, and DI.IF
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models across CIFAR-10 10, CIFAR- 100, ImageNet, and DVS-CIFARIO using the same network architecture and hyperparameters as those adopted in ( (Yao et al. 2022). while for DVS- ''stues as those adopted in (Fang et al. 202I). In addition, when comparing DLIF with the DH-LIF model, we follow the feedforward network architecture with comparable parameters and hyperparameter configurations provided in (Zheng al. 2024).
A.6.3 DATASETS
We conduct experiments custom-designed XOR task and several visual classification datasets, including CIFAR-10
al. 2009), (Krizhevsky et al. 2009), CIFAR-100 (Krizhevsky et al. 2009), ImageNet (Deng
(Amir al., 2017), and DVS-CIFARIO (Li al. 2017). Additionally,
evaluate performance on Atari games using the Gym (Brockman et al. 20166) and CleanRL (Huang
al. 2022 libraries.
XOR Task In the XOR task, each neuron receives 10 Poisson-distributed synaptic inputs over duration of 200 time steps, with Poisson rate of 0.5. For inputs where only half of the synapses activated, the target output spike count is 2. In contrast, for inputs where all synapses are activated, the target output spike count is A total of 1000 samples are generated for training and 200 for testing.
2D Correlated Input
We randomly sample two-dimensional Poisson inputs. each trial, both input dimensions share the same Poisson rate, which is a random number between 0 and 1. Each sequence has a duration of 1000 time steps, and we collect a total of 1200 samples, with 1000 used for training and 200 for testing.
We first fix two symmetric input second-order moment matrices, C1 and C2, such that ||C - C2ll = 1. The average number of input spikes is set to 200, with a sequence length of 1000. Based on the specified second-orde moments and firing rate, we then generate two distinct classes of inputs, with 500 samples per class for training and 100 samples per class for testing.
10D Correlated Input
CIFAR-10 The CIFAR-10 10 dataset comprises 60,000 natural images distributed across 10 classes, with 6,000 images per class. Of these, 50,000 images are designated for training and 10,000 for testing. The dataset is normalized scale pixel values to the range [0, I]. Subsequently, each image is replicated T times to generate temporal inputs, where T denotes the number of time steps in the evolution of the SNN.
CIFAR-100 The CIFAR-100 dataset is similar to CIFAR-10 but contains 100 classes of objects. It comprises 50,000 training samples and 10,000 test samples. The same preprocessing steps applied to CI are utilized for this dataset.
ImageNet The ImageNet dataset contains over million natural images across more than 20,000 classes, making it one of the largest and most diverse image classification datasets. In this work, we utilize the ImageNet- IK subset, which includes approximately 1.2 million training images and 50,000 validation images across 1,000 classes. Our data pre- pre-processing the usual practice, which randomly crops and flips the 224x224 image with general normalization method. Each image is then replicated T times to generate temporal inputs.
DVS-Gesture The DVS-Gesture dataset comprises 1,176 neuromorphic spiking gesture training samples and 288 test samples, each represented as 128x128 pixel frames. The dataset includes gesture types, such as waving, arm rotation, and forearm rolling. It is collected using a dynamic vision sensor, capturing data from 29 subjects under three distinct lighting conditions: natural light, fluorescent light, and LED.
DVS-CIFAR10
The -VS-CIFARIO dataset is derived from the original CIFAR- 10 dataset using neuromorphic vision sensor to generate temporal sequences. includes 9,000 training sequences and 1,000 test sequences.
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SHD and SSC Spiking Heidelberg digits (SHD) and spiking speech command (SSC) datasets convert the original audio data into the spike format through a bionic inner ear model. SHD contains about 10,000 high-quality recordings of English and German speech for digits ranging from 0 to 9. In particular, the SHD training and testing sets contain 8, 156 and 2,264 pieces of data, respectively; the SSC training. testing, and validation sets contain 75,466, 9,981, and 20,382 pieces of data.
The Atari game dataset comprises data generated from playing various Atari 2600 games. often used for reinforcement learning research. For this study, we evaluate model using 16 selected games: Atlantis, Beam Rider, Boxing, Breakout, Crazy Climber, Gopher. James Bond, Kangaroo, Krull, Name This Game, Pong, Road Runner, Space Invaders, Star Gunner, Tutankham, and Video Pinball.
Atari Games
A.6.4 TRAINING METHODS AND SNN ARCHITECTURES
XOR Task For the XOR task, we employ a single DLIF or LIF model. The time constant is set to 2. and the firing threshold is fixed at V4, = 1. We control the bilinear matrices to be symmetric with zero diagonals and initialize both the weights and bilinear coefficient from a normal distribution. Both DLIF and LIF neurons are trained using the BPTT algorithm, with the mean squared error as the loss function. Training is performed for 100 epochs with a batch size of 128 and a learning rate of 0.1. We constrain the Frobenius norm of K to be during training.
Network Correlation Simulation For the two-dimensional case, we train DLIF and LIF models to maximize the normalized correlation of the output spike trains toward 1. For the ten-dimensional case, we train the models such that the Frobenius norm of the difference between the output correlation matrices of the two input classes approaches 1. In both settings, the mean squared error is used as the constrained to be symmetric with zero diagonals, and both
loss function. The bilinear matrices the weights and bilinear coefficients are initialized from a normal distribution. Training is conducted using the BPTT algorithm for 100 epochs with batch size of 128 and learning rate of 0.1. During training, we constrain the Frobenius norms of both the weight vectors and the bilinear matrices to be 1.
My we consistently control the bilinear matrices to be symmetric with zero diagonals, and, except for the ablation experiments, we always fix the sparsity level at 90%. For all experiments, the training process is repeated five times with different initial values, and the mean and standard deviation of test accuracy calculated. For experiments where the origincal spiking neurons are replaced with DLIF neurons in SNNs, we adopt the same training methods and network architectures described in the original works, which are detailed as below:
Experiments in Large-scale SNNs In the results presented in Section 4
SLTT Spatial Learning Through Time (SLTT) (Meng et al. 2023 is training method designed to reduce the number of scalar multiplications and achieve memory efficiency that is independent of the total number of time steps, compared with BPTT. For CIFAR-10 10, CIFAR-100. DVS-Gesture, and DVS-CIFARIO, the adopted network architectures are ResNet- 18, R8sNet-18, VGG-11, and VGG-11, 1. respectively, with the total number of time steps set 6, 20, and 10, respectively. The time constant is set to 7 = 2. To improve compatibility with neuromorphic hardware, max-pooling layers are replaced with vverage-pooligg layers the network architectures. The loss function combines cross-entropy loss and squa---uuare--rroor loss (Deng al.] (002)). For all tasks, stochastic gradient descent (SGD) (Rumelhart et al. 1986) with momentum of 0.9 is employed to train the networks, and cosine annealing scheduler (Loshchilov & Hutter 2016) is used to adjust the learning rate. The training hyperparameters are as follows: for the CIFAR-IO 10 and CIFAR- 100 datasets, models are trained for 200 epochs with learning rate of 0.1, batch size of 128, and weight decays of 5 x 10 and 5 x 10-4, respectively. For the DVS-Gesture dataset, models are trained for 300 epochs with learning rate of 0.1, batch size of 16, and weight decay of 5 10 10-4. 4. For the DVS-CIFARIO dataset, models are trained for 300 epochs with learning rate of 0.05, batch size of 128, and weight decay of 10-4
ESG Evolutionary Surrogate Gradients (ESG) (Guo et al., 2022a) is a novel method for differen- tiable spike activity estimation, designed to ensure sufficient model updates during the initial stages
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of training and accurate gradient calculations at later stages. For CIFAR-10. CO0,R-100, and DVS- CIFARIO, 10, the adopted network architectures are VGG-16, VGG-16, and ResNet- respectively, with the total number of time steps set to 5, 5, and 10, respectively. The time constant is configured as т = 1.33. To enhance compatibility with neuromorphic hardware, max-pooling layers are replaced with average-pooling layers in the network architectures. Cross-entropy loss is used as the loss function. For all tasks, the SGD optimizer with momentum of 0.9 is employed to train the networks over 100 epochs, with batch size of 128, a learning rate of 0.1, and a weight decay of 10-4.
Online Training Through Time (OTTT) (Xiao et al. 2022) is an extension of BPTT that enables
OTTT
learning by tracking presynaptic activities and leveraging instantancous loss and gradients. The VGG-11 network architecture is utilized for all experiments on CIFAR- 10, CIFAR-100, DVS-CIFARIO, and DVS-Gesture, with the total number of time steps set to 6, 6, 20, and 10, respectively. The time constant is configured as 2. All models optimizer with a momentum of 0.9. For CIFAR-10, CIFAR-100, and DVS-CIFAR 10, models are trained for 300 epochs with batch size of 128. The initial learning rate is set to 0.1 and decayed to 0 using a cosine annealing scheduler. For DVS-CIFARI0 IIF, dropout is applied to all layers with a dropout rate of 0.1. The loss function combines cross-entropy loss and mean- rrean-squared-rroo loss. For DVS- Gesture, models are trained for 300 epochs with a batch size of 16 using the Adam optimizer. The initial learning rate set to 0.001 and decayed to O using a cosine annealing scheduler.
trained using the SGD
The threshold dependent batch normalization method based on propagation (STBP-tdBN) (Zheng et al. (02T)) addresses the gradient vanishing and explosion prob- lems while adjusting the firing rate. For CIFAR-10, ImageNet, Des-ure, and DVS-CIFARI 10, the adopted network architectures are ResNet- 19, ResNer-34, ResNet- 17, and ResNet- 19, respectively, with the total number of time steps set to 6, 6, 40, and 10, respectively. The time constant is configured as r = 1.33. Cross-entropy loss is used as the loss function. For all experiments, the SGD optimizer employed with an initial learning rate of 0.1, a momentum of 0.9, a batch size of 40, and 500 training epochs. The learning rate decays to 0.1r every 35 epochs.
STBP-tdBN
back-
TET
The Temporal Efficient Training (TET) algorithm (Deng et al, 2022) is designed to mitigate the loss of momentum in gradient descent with stochastic gradients (SG), facilitating convergence to flatter minima and improved generalizability. The ResNet-34 architecture is utilized for the ImageNet dataset, with the time constant configured and a weight decay of 4 x 10-5 is employed. The learning rate is initialized at 0. and decays to 0 using a cosine schedule. The network is trained for 120 epochs.
2. The SGD optimizer with momentum of 0.9
2024) is designed to achieve SSNN The Shrinking Spiking Neural Network (SSNN) (Ding et al.
low-latency neuromorphic object recognition. The VGG-9 network architecture utilized with a total of 8 time steps across all datasets. The time constant is configured as t = 2. All models are trained for 100 epochs with an initial learning rate of 0.1, which decays to one-tenth of its previous value every 30 epochs. The batch size is set to 64, and the SGD optimizer with momentum of 0.9 and a weight decay of x 10-3 is employed.
ESL The Error Compensation Learning (ESL) (Liu et al. 2025a) introduced learnable threshold clipping function, dual-threshold neurons, and an optimized membrane potential initialization strategy to mitigate the conversion error. For CIFAR-10 10 and ImageNet, the adopted network architectures are 18 18 and VGG-16, with the total number of time steps set to 64 and 128. respectively. We use the same training strategy as in STBP-tdBN since the original text does not provide a detailed description of the training parameters.
TSER Temporal Separation with Entropy Regularization (TSER) ( Yu et al. 2025a) introduced knowledge distillation in spiking neural networks. The teacher models for CIFAR-IO, CIFAR and ImageNet datasets are VGG-16, VGG-16 16 and ResNet-34, respectively. The time steps are all set to 4. We use the same training strategy as in STBP-ddBN since the original text does not provide a detailed description of the training parameters.
ESTA Frequency-based Tempooal Attention (FSTA) module is proposed to enhance feature learning in SNNs (Yu et al. 2025b). For CIFAR-10, CIFAR-100, ImageNet, and DVS-CIFAR1O, the
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adopted network architectures ResNet-19, 19, ResNet-19, ResNet-34, and ResNet-20, respectively, with the total number strategy in STBP-tdBN since the original text does not provide detailed description of the training parameters.
of time steps set to 2, 2, 4, and 16, respectively. We use the same training
SSSA Saccadic Spike Self-Attention (SSSA) method is proposed to address the issues of the mismatch between the vanilla self-attention mechanism and
spike trains (Wang et al.) 2025b). ViTs are used for CIFAR-I0, CIFAR-IOK ImageNet and DVS-CIFAR 10 datasets with time steps of 4. We use the same training strategy as in STBP-tdBN since the original text does not provide a detailed description of the training parameters.
Spike-driven Transformer The Spike-drived Transformer is SNN architecture that incorporates the spike-driven paradigm into Transformer (Yao et al. 2023a). This architecture combines the low power of SNN and the excellent accuracy of the Transformer. The time constant is set For ImageNet, the batch size is set to 256 during 310 training epochs with a cosine-decay learning rate whose initial value is 0,0005. The optimizer is Lamb and the timestep is T other four datasets, we employ timesteps 7 = 4 on CIFAR-I0 and CIFAR-100, and T = 16 on DVS-CIFAR 10 and DVS-Gesture The training epoch for these four datasets is 200. The batch size 32 for CIFAR 1/100 and 16 for DVS-Gesture/CIFAR 10. The learning rate is initialized to 0.0005 for CIFAR 10/ 100, 0.0003 for DVS-Gesture, and 0.01 for DVS-CIFAR I0. All of them are reduced with cosine decay. In addition, the network structures used in CIFAR-10, CIFAR-100, ImageNet, DVS-CIFARIO, 10, and DVS-Gesture are: Transformer-2-512 Transformer-2-512. Transformer-100 10-512 Transformer-2-2555 and Transformer-2-222 respectively, where Transformer-L-D in represents model with L encoder blocks and D channels.
2.
For the
Neta-Sri Meta-SpikeFormmrr is general Transformer-based SNN architecture for future next- texerrennnttinn Transformer-basee neuromorphic chip designs (aa et al. 2024 A 55M Transformer used for ImageNet dataset with 4 timesteps. The AdamW employed with an initial learning rate of Le - 3 that will decay in the polynomial decay schedule with a power of 0.9. To speed up training, we warm up the model for 1.5k iterations with linear decay schedule.
DSQN The network architecture of Deep Spiking Q-Network (DSQN) (Chen et al. 2024) is struc- tured as Input-32C8S4-SN (0--844-SN-64444222 64C442-SSM SN-22-NN-66 B4C331-SN-FFaatee F552ttnn-511-S5 -2-SN-NN-SSM where SN represents spiking neurons, which can be either LIF DLIF neurons, and NA denotes the number of actions in the task. The model is trained over a total of 20 million frames. During evaluation, the agent begins each episode with random number (up 30) of no-op actions, and the behavior policy follows an e-greedy approach, with fixed at 0.05. The simulation timesteps are set to 8, and the membrane time 2. The model uses a minibatch size of 32, replay start size of 50,000,
constant is configured and a replay memory size of 1,000,000. The target network is updated every 10,000 steps. The Adam optimizer is employed with a learning rate 0.00025 and an value of X 10 Exploration starts at an initial value of 1.0, decaying linearly to a final value of 0.1 over 1,000,000 frames. The
maximum number of no-op actions per episode is set to 30. These hyperparameters remain consistent across all games.
All implementations are built on the PyTorch (Paszke et al. 2019) and SpikingJelly (Fang et al., 2023) frameworks. All the experiments are conducted on an NVIDIA Tesla A100 GPU with 6,912
CUDA cores and 432 tensor cores.
27


Reviews:
[{"review_id": "Eny2AbiTa4", "reviewer": "Reviewer_9rV7", "summary": "This study proposed a new spiking neuron model called DLIF, which can conduct dendritic computations.  Experimental results show that this neural model achieves SOTA performance on five computer vision datasets", "questions": "1. Section 4 \"Specifically, we set the sparsity level to 90%, a\nchoice that is both biologically inspired and empirically validated by an ablation study, as shown in\nSection 4.4.\". Please explain which dimension the sparsity is applied and how it is controlled before settingthe  sparsity level.\n\n2. Is there any open-source plan for this study?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. The proposed neuron model is biologically-inspired and the single neuron theorem is solid.\n2. The experiments on XOR verified the theory.\n3. Results are validated on several datasets", "weaknesses": "1. The energy metrics are estimated using numbers from a paper from a decade ago. Further justifications should be provided, and the energy benchmarking setting should be clearly presented, e.g., which GPUs are the target, how SNNs are set up and run on such GPUs, and how many GPU it required.\n2. The paper compared many SNN algorithms and reproduced their results. It can be beneficial to provide codes (Assuming the ability to replace any neuronal models on any SNN algorithm is one of the main contributions of this paper), instead of just listing the experimental settings in the Supplementary materials for each SNN algorithm. It can make other researchers much easier to follow this study and develop a new spiking neuronal model on top of this study.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uDVEaLCLrI", "reviewer": "Reviewer_Uc9p", "summary": "The paper proposes the Dendritic LIF (DLIF) model that extends conventional LIF by incorporating bilinear dendritic integration. Through theoretical analysis and empirical simulations, the authors show that a single DLIF neuron can effectively capture correlations in input spike trains. Experiments across diverse deep SNN architectures and datasets demonstrate that DLIF can achieve SOTA performance, highlighting its scalability.", "questions": "See “Weakness” for major questions. Below are some additional minor questions and suggestions:\n\n1.More details should be provided regarding the computation of theoretical energy cost. The explanation in Appendix cannot address my key concern: how is the energy consumption of a bilinear operation calculated?\n\n2.The parameter counts in Table 1 and 2 are weird for me. For a synapse layer with $N$ presynaptic neurons and $M$ postsynaptic neurons, the linear weight is a $M\\times N$ matrix, while the bilinear weight is a $M\\times N \\times N$ tensor, as stated in Line 192 of the manuscript. Even with a 90% sparsity constraint and an all-zero diagonal, there should still be about $0.1\\times M \\times (N^2-N)$ more learnable parameters, which is much larger than $M \\times N$. However, the reported parameter increase is only about 10%. Please clarify in detail how the parameter counts are computed.\n\n3.The authors report the results of DLIF networks trained with OTTT. However, OTTT is an eligibility-trace-based online learning method designed for optimizing $\\mathbf{W}$, the linear weights. Its extension to bilinear weights is not trivial. Please explain how to use OTTT to optimize bilinear weights.\n\n4.Typos: “biliear” in Related Work should be bilinear; “as a binary” in Page 3, Line 159 should be “as a binary matrix”.\n\n5.Citation formatting: please use `\\citep{}` instead of `(\\cite{})`. The displayed text should appear as (xxx et al., 20xx) instead of (xxx et al., (20xx)).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.Introducing dendritic computation to large-scale neuromorphic computing is an advanced and compelling research topic.\n\n2.The paper is clearly structured (from single neuron to large-scale SNNs) and is generally well written. The mathematical derivations are rigorous. The figures and tables effectively convey the key ideas and results.\n\n3.Experiments cover a wide range of tasks and SNN architectures. DLIF consistently outperforms LIF on (almost) all the tasks, showcasing the method’s effectiveness and generalizability.", "weaknesses": "1.The novelty of the proposed model is not clearly stated. What are the key differences or innovations of DLIF with the model proposed by (Li et al., 2019)? \n\n2.Although comparisons are made with point neuron models like PLIF, GLIF, QIF and EIF (Table 3), no comparison is made with dendritic or multi-compartment spiking neurons like (Zheng et al., 2024). The authors claim that DLIF is inspired by dendritic computation, so comparing it with other dendritic neuron models would strengthen the work’s soundness.\n\n3.While inference cost metrics like FLOPS and theoretical energy consumption are provided, there is no information regarding training cost. How much extra training time and memory overhead does the bilinear operation introduce? A comparison with LIF or other spiking neuron models should be made.\n\nLi, Songting, et al. \"Dendritic computations captured by an effective point neuron model.\" Proceedings of the National Academy of Sciences 116.30 (2019): 15244-15252.\n\nZheng, Hanle, et al. \"Temporal dendritic heterogeneity incorporated with spiking neural networks for learning multi-timescale dynamics.\" Nature Communications 15.1 (2024): 277.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PS07T0ZRGf", "reviewer": "Reviewer_KEoS", "summary": "The paper proposes a DLIF neuron that augments the input current with a bilinear dendritic term ($s^\\top K s$), argues theoretically that DLIF captures pairwise input correlations (single-neuron) and preserves/propagates correlation structure (network-level), and reports consistent gains across CIFAR-10/100, ImageNet, DVS-Gesture, DVS-CIFAR10, and Atari RL with small overheads. The idea is timely and simple to integrate into existing SNN stacks. Results are encouraging, but the claims about complexity/energy, the parameterization and sparsity of ($K$), and the scope/limits of the theory need clearer, checkable evidence.", "questions": "1. How about replace sparse matrix to low-rank matrix. It can better reduce parameter number and computational cost.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- DLIF replaces linear current summation with an additive bilinear term (s^\\top K s); formulation is simple, but performance is good. \n- Static and neuromorphic vision, multiple architectures (VGG/ResNet/Transformer), and training regimes (SLTT, OTTT, TET, STBP-tdBN, etc.) show consistent improvements with small reported overheads.", "weaknesses": "1. The text states ($K$) reflects spatial relationships and is intensity-independent, yet in networks it is learned. Are there structured priors (locality, block-diagonal, cross-channel only) or sharing tied to receptive fields? A comparison of fully learnable ($K$) vs. structured/low-rank/near-neighbor ($K$) would clarify the interpretability–performance–cost trade-off. \n\n2. Since $K \\in \\mathbb{R}^{N\\times I \\times I}$ (although sparse), computational cost, parameter accounting, and training speed need to be better compared.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "2HhjPNzIg3", "reviewer": "Reviewer_VTNi", "summary": "This paper introduces the DLIF neuron—a variant of the Leaky-Integrate-and-Fire model that implements dendritic bilinear integration. Compared with the standard LIF, DLIF adds a bilinear dendritic term that can be evaluated with minimal overhead, requiring only AND operations. The authors theoretically prove that a single DLIF unit can discriminate between spike trains with identical mean firing rates but different correlation structures, and that a two-layer DLIF-based SNN preserves input correlation structures more effectively than its LIF counterpart. Extensive experiments on multiple datasets and across many existing architectures or training methods show consistent accuracy gains when LIF neurons are replaced by DLIF.", "questions": "The authors claim that DLIF neurons incur “no significant increase in computational cost”; however, in my experience introducing stateful synapses into an SNN usually raises training expenses dramatically. Could the authors supply a direct comparison of training/inference time and memory consumption between DLIF and vanilla LIF neurons?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Theoretical analysis rigorously supports the advantages of DLIF.\n- Exhaustive empirical evaluation: DLIF consistently improves performance when plugged into a wide range of models.", "weaknesses": "- DLIF appears difficult to realize at the neuron level alone, because it requires storing the presynaptic spike vectors from the previous layer rather than just the aggregated synaptic current. Implementing this seemingly demands stateful synapses, potentially increasing both training time and memory usage.\n- The proposed method was evaluated on multiple datasets by replacing the LIF neurons in prior works with DLIF neurons. Tables 1 and 2 show consistent and stable improvements, a result that is highly encouraging and suggests DLIF could universally outperform LIF and become a new standard neuron model for SNNs. However, the absence of released source code and experiment logs reduces credibility and prevents readers from examining implementation details.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)