Fractional-Order Spiking Neural Network
Authors: Chengjie Ge
Yufeng Peng
Zihao Li
Qiyu Kang
Xueyang Fu
Xuhao Li
Qixin Zhang
Ren Junhao
Zheng-Jun Zha
Venue: ICLR 2026 Poster (2026)
Keywords: spiking neural networks
fractional order differential equations
Primary area: applications to computer vision, audio, language, and other modalities

Abstract: Spiking Neural Networks (SNNs) draw inspiration from biological neurons to enable brain-like computation, demonstrating effectiveness in processing temporal information with energy efficiency and biological realism.
Most existing SNNs are based on neural dynamics such as the (leaky) integrate-and-fire (IF/LIF) models, which are described by *first-order* ordinary differential equations (ODEs) with Markovian characteristics. This means the potential state at any time depends solely on its immediate past value, potentially limiting network expressiveness.
Empirical studies of real neurons, however, reveal long-range correlations and fractal dendritic structures, suggesting non-Markovian behavior better modeled by *fractional-order* ODEs.
Motivated by this, we propose a *fractional-order* spiking neural network (f-SNN) framework that strictly generalizes integer-order SNNs and captures long-term dependencies in membrane potential and spike trains via fractional dynamics, enabling richer temporal patterns. We also release an open-source toolbox to support the f-SNN framework, applicable to diverse architectures and real-world tasks.
Experimentally, fractional adaptations of established SNNs into the f-SNN framework achieve superior accuracy, comparable energy efficiency, and improved robustness to noise, underscoring the promise of f-SNNs as an effective extension of traditional SNNs.

Published as a conference paper at ICLR 2026
FRACTIONAL-ORDER SPIKING NEURAL NETWORK
Chengjie Gel; Yufeng Peng', Zihao Li', Qiyu Kang" 7 Xueyang Fu', Xuhao Li', Qixin Zhang',
Junhao Ren', Zheng-Jun Zha'
Universityy of Science and Technology of China Anhui University
Nanyanng Technological University
com www.//hysGT//ppikelllelt
ABSTRACT
Spiking Neural Networks (SNNs) draw inspiration from biological neurons enable brain-like computation, demonstrating effectiveness in processing temporal information with energy efficiency and biological realism. Most existing SNNs are based on neural dynamics such as the (leaky) integrate-and which are described by first-order ordinary differential equations (ODEs) with Markovian characteristics. This means the potential state at any time depends solely on its immediate past value, potentially limiting network expressiveness. Empirical studies of real neurons, however, reveal long-range correlations and fractal dendritic structures, suggesting non-Markovian behavior better modeled by fractional ODEs. Motivated by this, we propose a fractional--rree -order spiking neural network (f-SNN) framework that strictly generalizes integer-order SNNs and captures long- term dependencies in membrane potential and spike trains via fractional dynamics, enabling richer temporal patterns. We further release an open-source toolbox, spikeDE, to support the f-SNN framework across diverse architectures and real- world tasks. Experimentally, fractional adaptations of established SNNs into the f-SNN framework achieve superior accuracy, comparable energy efficiency, and improved robustness to noise, underscoring the promise of f-SNNs as an effective extension of traditional SNNs.
(IF/LIF) models,
INTRODUCTION
Neural networks have evolved substantially as researchers continuously explore models that better reflect biological neural systems while maintaining strong performance. Traditional artificial neural networks (ANNs) excel across many tasks (Krizhevsky al., 2012; LeCun al., 2015; Vaswani et al., 2017) but differ from real biological mechanisms, and modern models require far more compute than the human brain (Dhar, 2020). This gap has motivated Spiking Neural Networks (SNNs) (Maass, 1997; Ghosh-Dastidat & Adeli, 2009; Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021; Zhou et al., 2022), which model neural activity more realistically by communicating through discrete spikes rather than continuous values. Their event-driven computation paradigm allows for significant energy savings, particularly when implemented on neuromorphic hardware (Roy et al., 2019; Pei et al., 2019). Additionally, SNNs naturally handle time as part of their processing, making them well-suited for tasks with time-series data or real-time interactions in changing environments (Yao et al., 2023b; Luo al., 2024; Yao et al., 2021). These features make SNNs strong candidates for applications that need both energy efficiency and good temporal processing.
Despite these advantages, existing SNN models predominantly describe spiking neuronal membrane- potential dynamics using the widely adopted integrate-and-Fiig (IF) and Leaky Integrate-and santte-and-Frre (LIIF) neurons (Stein, 1967), along with variants including nonlinear spike initiation (Ermentrout, 1996; Fourcaud-Troeee constants (Koch et al., 1996; Zhang et al., 2025), and threshold adaptation or learning (Bellec et al., 2018; Benda, 2021). These models discretize first-order ordinary differential equations (ODEs) which contains only d/dt terms (Hodgkin & Huxley, 1952; Maass, 1997; Ghosh-Dastie & Adeli, 2009; Eshraghian al., 2023b) and assume Markovian property in which the current state depends
et al., 2003), ternary spikes (Guo et al., 2024), adaptive membrane time
"First two authors contributed equally. 1 "ooresponndecee Qiyu Kang <qiyukang@ustc.ddu.cns

---

Published as a conference paper at ICLR 2026
mainly on the immediate previous state (see (9)). While this simplification enables computational tractability, it fundamentally limits the expressiveness of these networks. Neurophysiological research has demonstrated that real neurons display far more complex behaviors influenced by long-term correlations (Gilboa et al., 2005), fractal dendritic structures (Coop et al., 2010; Kirch & Gollo, 2020), and the interaction of multiple active membrane conductances (La Camera al., 2006; Miller & Troyer, 2002). These dynamics cannot be adequately captured by integer-order models (Ulanovsky al., 2004; La Camera et al., 2006; Miller & Troyer, 2002; Spain et al., 1991) and suggest that non-Markovian dynamics play significant role in biological neural computation. Fractional calculus instead offers mathematical tools for modeling such dynamics better than standard first-order ODEs (Diethelm, 2010; Baleanu et al., 2012). In contrast to integer-order calculus, the fractionul-order derivative d /d1", with non-integer a values, considers the entire history of a function, weighted by power-law kernel. The fractional leaky integrate-and-ff studied in (Teka et al., 2014; Deng et al., 2022), serves as an example of applying these concepts. This model can effectively explain spiking frequency adaptations observed in most biological neurons (Ha & Cheong, 2017) and has been shown to generate more reliable spike patterns than integer-order models when subjected to noisy input (Teka et al., 2014). Despite these promising findings, the integration of SNNs and fractional neurons remains largely unexplored area (Lee & Monahan).
(f-LIF) neuron dynamic, introduced and
Pyramidal
Neurons
Synaptic Membrane Potential
Integer-Order RC Circuit Model
Leaky Integrate-and-
Fire Model
Cerebral Cortex
Occipital Lobe
Cerebellum
Brainstem
Soma Membrane
Fractional-Order RC Circuit Model
Fractional-Leaky Integrate- and-Fire Model
potential
Figure 1: Comparison of traditional SNN and f-SNN framework.
In this paper, we introduce a generalized fractional-order SNN (f-SNN) framework, which incor- porates fractional-order dynamics into the neuronal membrane potential charging. By replacing the first-order ODE neurons traditionally used in SNNs with fractional -orden ODEs ff naturally captures long-tern dependencies that are beyond the capability of standard SNN models, leading to improved performance on tasks that require complex temporal processing. We highlight that framework is more general framework which subsumes many traditional SNNs special instances by setting a 1. We evaluate foatteef-a fSNN models on multiple benchmark datasets spanning neuromorphic event-driven vision, graph domains, and static vision fields. Experimental results show that f-SNN models consistently outperform conventional SNN models across various evaluation metries. Moreover, f-DDES are robust to perturbations (Sabatier et al., 2015; Kang al., 2024c); in particular, neural f-ODEs admit tighter imput-outpr perturbation bounds than integer-order models (Kang et al., 2024c). Building on this, an additional advantage of our proposed f-NN FSNN framework its superior robustness under input perturbations. These findings underscore the practical advantages of integrating fractional-ordes dynamics into SNNs and point to the broader applicability of our f-SNN in real-world scenarios.
Main contributions. Our objective in this paper is to formulate a generalized fractional-order SNN framework. Our key contributions are summarized follows:
We propose an f-SNN framework that integrates f f-ODEs into SNNs to naturally capture long- term dependencies using the fractional orderr operator d² /dt". This framework generalizes the traditional class of integer-order SNNs that use IF, LIF neuron dynamics, and their variants, subsuming them special case by setting a
2

---

Published as a conference paper at ICLR 2026
We establish fundamental theoretical distinctions f-SNSs and traditional SNNs, proving that fractional-ordes dynamics confer three key advantages: persistent memory through power-law relaxation, irreducibility to finite classical ensembles, and enhanced robustness to perturbations. We underscore the compatibility of F-SN, emphasizing its ability to be seamlessly integrated to augment the performance of many existing SNNs by using non-integer a with various neural network architectures like convolutional neural networks (CNN). Transformer, ResNet, and mul- tilayer perceptron (MLP) (Vaswani et al., 2017: 7; LeCun et al., 1989; He et al., 2016; Zhou et al., 2022). We provide the community with open-source, out-of-the-boo toolbox to support the f-SNN framework (see supplementary code and Section E). We conduct extensive experiments on multiple datasets, demonstrating that consistently improves traditional SNNs, achieving superior accuracy, comparable energy efficiency, and enhanced robustness.
2 PRELIMINARIES
This section reviews essential concepts. We introduce fractional calculus, which generalizes deriva- tives non-integer orders and naturally models systems with memory or non-local dependencies. We then outline conventional SNN approaches based on discretizing integer-order neuron dynamics.
2.1 FRACTIONAL CALCULUS
When examining a function y(t) with respect (w.r.t)) time t, we traditionally define the first-order derivative as the instantaneous rate of change: dt various definitions of fractional derivatives (Tarasov, 2011). We focus on the Caputo fractional derivative D for the formal definition of d /dt" (Diethelm, 2010), which has the notable advantage of allowing initial conditions be specified in the same manner as integer-order differential equations.
dy(t) == lim A(t+At)-w(t) The literature offers
At
Definition 1 (Caputo Fractional Derivative). For function v(() defined over interval [0, T), its Caputo fractional derivative of order a E (0, 1] is given by (Diethelm, 2010):
1 D² = T(1 a) S (1 (1- (1- (T)
where y (T) denotes the first-order derivative of y(T).
Remark 1. (1) reveals that the fractional derivative incorporates the historical states of the function through power-weighted integral term when a E (0,1), highlighting its memory dependence.
As + 1, the Caputo derivative D converges to the standard first-order derivative d àe- Indeed, letting F(s) = [t) be Laplace transform of (t), we have L ( f(t)] = 8 F(() -8a -0011s00000 (Diethelm, 2010)[Theorem 7.1]. As a a-> 1, the Laplace transform of the Caputo fractional derivative converges to that of the traditional first-order derivative sF(s) (0). Consequently, for a
D'y = y', uniquely determined via the inverse Laplace transform (Cohen, 2007).
first-order ODE and its fractional extension with Caputo derivative can be written as
integer-order ODE: dy(t) dt f(t v(t)): fractional-order ODE: Dd (()()) (t. v()):
(2)
where f defines the system dynamics and initial condition (0) = is specified in both cases.
2.2 EYTEGER-ORDEL SPIKING NEURON AND SNN
Existing SNN models predominantly describe spiking neuronal membrane- potentiaa dynamics using discretized first-order ODEs with derivative d/dk dt, including the widely adopted IF and LIF dynamics (Stein, 1967) and variants with adaptive membrane time constants or threshold adaptation/lcarning (Koch al., 1996; Zhang et al., 2025; Bellec et al., 2018: Benda, 2021). We present only standard IF and LIF in the main paper as a showcase; however, SNNs based on other neuron variants can be encapsulated and extended within f-SNN framework.
IF and LIF neurons. Let (t) denote the membrane potential, Im (t) the input current, R > the membrane resistance, and the membrane time constant. The standard subthreshold dynamics
3

---

Published as a conference paper at ICLR 2026
of IF and LIF are described by the following first-order ODEs:
IF neuron dynamics: dt(() R (n))s,
(4)
dt
dU
LIF neuron dynamics: dt U(t) + R.RI (t). (m)
(5)
spike S(t) is emitted when (tt)) crosses the threshold 0, i... S(1) = H (U(())-a 0), where (-)
denotes the Heaviside step function. Upon spiking, uses either soft reset hard reser:
1) soft reset: U((+)= - U(t)) 0; 2) hard reset: U U((+) + Urseet
(6)
Traditional SNN based on standard IF and LIF neuron dynamics. Many SNNs are based on the
neuron dynamics described in (4) and (5). In the simplest case, the forward Euler method is employed
to solve a first-order ODE (2). Let h >> 0 be the discretization step size, tx = kh, N T/h, and let ys denote the numerical approximation of y( We have
Wk+1 yk + h ftz, yk), k IN
(7)
To make this time-varying solution compatible with sequence-based neural network models, discretize time and treat k: as the sequence index. Correspondingly, applying (7) to (4) and (5) yields
hR
h Tin,k: LIF (discrete): Uk+1 -1-
hR
IF (discrete): U2+1 UR
Im
(8)
T
In practice, factor is often absorbed into learnable synaptic weights, and the input current represented as x*) where x-(*) the presynaptic spike vector or feature map (e.g., produced by Convolution, MLP, ResNet, or Transformer) with & denoting the learnable synaptic weights of those layers. For simplicity, in the following omit & and denote it simply by Xę. For computational
efficiency, we adopt the common simplifications = and R 1. and define 3 = Together with spiking and reset mechanisms, have the following iterations:
IF charge: UR = Uk-1 Xx.
or LIF charge: Us -U--1+ Xx.
spike: S = H(UR 0)
reset: (soft) Us + Us 0S, or (hard) Us + (1 - (1) Us + Sv Ureset
Spikes are discrete and con-differentiable, which complicates SNN training. The surrogate croggate-gradien method (Wu et al., 2018) keeps the hard spike H - #) in the forward pass but uses a smooth surrogate for its derivative backpropagation. common choice is threshold-shifted sigmoid, H(U 0) 0) R a(U) I+2--0 which preserves discrete firing while enabling gradient flow.
a= 0.2
Input
Input
0.8
0.4
+0.25
+0.16
+0.45
+0.25
+0.15
0.6
66
a 0.8
U10
(U()))
Spikal Spike!
1.0
Vth -
Whtl memory
8
5/4
0.25
8:38
0.27
mamerg
neriory
0.30
52
Output Spike
None!
Output Spike
60
2
JLIF
Figure 3: Mittag-Leffler function Eg (-t%) For LIF shows fast exponential decay
Figure 2: SNN FSNN dynamics. In f-SNN,,, past brane potentials influence the current state via a power-law (Ea()t) = e e-x; for 0 < o < 1, exhibits slow algebraic decay, reflecting memory.
memory kernel; traditional integer-order SNNs lack this.
3 f-SNN FRAMEWORK
We present the -SNN framework in this section using fractional spiking neuronal dynamics based on f-ODEs, which generalize integer-order neuron dynamics such standard IF and LIF neurons (4)
4

---

Published as a conference paper at ICLR 2026
and (5). To make the time- sere-iarying solution compatible with neural network models, follow the procedure in Section 2.2 to discretize time and enable iterations. Section 3.2 reveals the fundamental distinctions between f-SNNs and traditional SNNs through analysis of their long-time behavior, demonstrating how f-SNNs provide persistent memory via power-law relaxation, irreducibility to finite classical ensembles, and enhanced robustness.
3.1 FRAMEWORK
Traditional integer-order SNNs, as discussed in Section 2.2, model subthreshold spiking neuronal dynamics with first-order ODEs; (4) and (5) are representative examples. In our general f-SNN framework, we replace the first-order derivative d/dt with the generalized Caputo fractional derivative D of order E (0, 1]. Since IF and LIF are the dominant neuron models used in traditional SNNs, we present only their fractional extensions in the main paper as a showcase; however, many other neuron variants can likewise be encapsulated and extended within f-NN framework. We begin with the presentation of f-IF and andf-LIF neurons.
f-IF f-LIF neurons. The fractional dynamics of IF and LIF are described by the fODEss:
f-IF neuron dynamics: T (t = RI (m)(t), f-LIF neuron dynamics: DC) - U() + (1)
(10)
(11)
Spike generation and reset follow the same rules as in the integer-order case: S(t) H (U
These dynamics naturally introduce a memory effect: the current membrane potential depends on the entire history of the potential because, by definition (see (I)), the Caputo derivative includes an integral over past states. Biologically, such modeling is consistent with observed milke-frequency adaptation and long- memory behaviors (Teka et al., 2014; Ha & Cheong, 2017). The order a controls 1 recovers the standard IF/LIF models, while
the degree of adaptation- law memory and increased temporal correlations in the potential trace. Fractional neuron models are also observed to produce reliable spike patterns under noisy input (Baker al., 2024).
induces power-
f-SNN based on f-IF and f-LIF neuron dynamics: In Section 2.2, we apply the forward Euler method discretize standard IF/LIF dynamics and obtain integer-order SNNs. Here, the f-IF (10) f- (11) neurons exhibit fractional dynamics that belong to the f-ODE class (3). We instead use the fractional -ashsn-Bsshfoth--a to achieve this goal. Using the same time grid as above, tg kh with N = T/h and step size h > 0, and letting yk denote the numerical approximation of y (t&), we obtain
(ABM) predictor discretization (Diethelm al., 2004)
V/4 = yuu T(a)
Ms f(1;:1g), k = 0, 0.........A -
(12)
where the weight coefficients are μ3, = A (k-))=)))) This formulation makes the memory effect explicit by incorporating weighted contributions from all past function evaluations, reflecting the nonlocal nature D When a = 1, (12) reduces exactly to the Euler method (7), further highlighting the compatibility between the SSNN framework and traditional SNNs.
Applying (12) to (10) and (11) yields the fractional discrete updates:
k-1 Uj.k Jin.j
f-IF (discrete): U2 = Up +
T
T(a)
1=
k-1
f-LIF (discrete): Up Uo
Σ ==
Ug.k ( U, + R Rlmss)
T [(a))
Similar to Section 2.2, we denote the general input as Xg. where X₃ is the presynaptic spike vector or feature map produced by various architectures (convolution. MLP, ResNet, Transformer, etc.). For simplicity, we set h = 1 and R = Note that 3 = 1 11-- & in IF/LIF neurons does not apply to the fractional cases. Instead, obtains history-convolution with stationary pawer-law kernel. Define
5

---

Published as a conference paper at ICLR 2026
(0) Om 7. [لل [Im 1) m²]. Then the fractional iterations (charge-spike-reset) follows:
f-IF charge: Us Ug Σ
k-1
or f-LIF charge: Uv = Un +
Σ
(a) (-Uk-I-m -x-m)).
(13)
m=0
spike: Sy: H(U 8)
reset: (soft) Us- +- UR- 0& or (hard) UR +- (1 Sx)UR + S4
Here, spiking and reset are applied at each step as usual. We follow the literature to use the surrogate- gradient method (Wu al., 2018) train f f-SNN that keeps the hard spike HUU - 0) in the forward pass but uses a smooth surrogate for its derivative in backpropagation.
Remark 2. Note that cia) is causal and decays a power law, explicitly encoding memory. The profile of Cm visualized in Fig. 8, highlighting the algebraic decay characteristic of fractional order systems. When + 1, we have (1) 1/T for all (a constant kernel), and taking first differences of (13) recovers the Euler recursions (9). For efficiency, we may leverage the short- memory approximation principle (Deng, 2007; Podlubny, 1999) and truncate the sum in (13) to mm= k-My iee, a sliding memory window of fixed width M. With fast (FFT-based) convolution, the full-memory case can be computed O(N log N) time (Mathieu et al., 2013), while the truncated window yields ONN The full model complexity is summarized in Section C.S.
3.2 THEORETICAL ANALYSIS
In this section, we theoretically distinguish the f-SNN framework from traditional SNNs. We begin by proving that f-SNNs exhibit a persistent memory effect characterized by genuine long-range temporal dependence. We then demonstrate that the dynamics f-SNNs generally cannot be exactly realized by any dintesiiinniionll linear system of integer-order modes, thereby establishing that fractional-order systems strictly exceed the expressive capacity of integer-order models. Finally, we prove that S-NNNs demonstrate superior robustness input perturbations.
We first analyze membrane- neembrane-poeentall relaxation under constant input, showing how distant past inputs keep influencing the present. For intuition, we focus on the LIF and f-LIF neurons and use the continuous formulations (5) and (11):
Proposition 1 (Long-ment) memory Behavior). Under a constant current input (t) = le, assume the input is small enough that no spiking occurs over the interval considered (subthreshold regime). Then the solutions to the LIF (5) and the f-LIF dynamics (11) are
ULIP (t) Rle + [Uo Rle] e t/T,
(14)
UIFLIF (t) Rlee + (Un Rlc) Ed(-t(/T).
(15)
* respectively, where Ea Eb=o F(aktij is the Mittag-Leffler function Diethelm (2010). Key properties include:
When Ei(-t/T) = t/T,, which is the classical exponential relaxation (Eshraghian et al., 2023a)[Eq(211. For < a < 1, E, exhibits (i) initial stretched decay and (ii) a power-law tail for large t:
7
E.
r(1 a) t 00,
These behaviors are visualized in Fig. 3.
Remark 3. While both LIF and f-LIF converge to the same steady state, Proposition highlights fundamentally different relaxation behaviors. The LIF uses an integer-order derivative (Markovian dynamics: future evolution depends only on the current state) and shows exponential relaxation e t/T, characteristic of memoryless processes. In contrast, the f-L.F employs the fractional derivative

---

Published as a conference paper at ICLR 2026
D which is inherently non-Markovian, incorporating power-weighted integral over the entire past history. This is reflected in the Siitag-Lefflee relaxation Ea (a(--//T), for 0 < a < 1, its power-law tail (~ t=) indicates that past inputs decay algebraically slow rather than exponentially fast, creating a persistenl memory influence. This slow decay means that imputs from the distant past continue to influence the current membrane potential, enabling the f-L.F to naturally capture long-term temporal correlations.
The f-SNN framework demonstrates superior robustness compared to traditional SNNs. Empirical studies show f-LFF neurons maintain reliable spike patterns under noisy imputs (Teka al., 2014). Here, we provide theoretical robustness guarantees.
Theorem 1 (Robustness of f-SNN). Consider fructional f-IF neuron governed by the dynamics t (L) = (l) with fractional order < a 1 and initial condition Uo = 0. Under additive perturbation (where |e] < Lc), the system exhibits the following robustness properties relative to the classical integer-order model (a 1):
constant input current Le subject
Membrane Potential Robustness: The membrane potential deviation due to perturbation evolves as:
Re -I - 1)²
(F)
to (sub-linear growth)
(16)
Re
A (t) = -t (linear growth)
(17)
For 0 < 1, the fractional-order dynamics suppress long-term perturbation accumulation through sub-linear temporal scaling.
Spike Timing Sensitivity: For small perturbations < Ta. the spike time shift magnitude scales as:
A x (1(41/a)
(18)
A 16-
(19)
Since (1 + > 2 for 0 < a a<< 1, the fractional-order model exhibits enhanced spike timing robustness for high input currents.
Remark 4. The fractional-ories dynamics yield distinct robustness advantages. The sub-linear perturbation growth t (a < 1) significantly suppresses long-term accumulation compared to linear growth in classical models. Additionally, the enhanced spike timing stability becomes crucial for precise temporal coding applications (Bohte al., 2002: Booij & tat Nguyen, 2005; Rathi al., 2019). These properties make f-SNNs particularly suited for tasks requiring sustained accuracy and temporal precision under varying imput conditions, as conftrmed by our experiments in Section 4.
We now establish that f-SNNs possess computational capabilities that fundamentally exceed those of finite integer-order systems:
Theorem 2 (Irreducibility of f-IF Dynamics to Finite Classic LIF Ensembles). Let US-IF denote the W
trajectory of f-FF neuron with order (0, There exist no finite integer W, weights {Ø; {0i)e-1 W and leak factors ( such that the following holds:
Úx = Σ
0.U (IF((.))
U. Vk.
for general input Xg. The impulse response error of the approximation is O(ka-)), decaying algebraically slowly. The f-IF neuron is mathematically equivalent to an aggregate of integer-order LIF neurons if and only if W -> oo, specifically as an integral over continuum of leak factors.
Remark (Implications for Expressive Power). A single f-IF neuron represents a continuum of timescales that would require infinitely many integer-order SNN units for exact equivalence. The slow O(ka-1) error decay confirms that such long-range dependencies are inaccessible to fnite-order models. Moreover, this expressivity advantage is not "washed out" " by the spiking nonlinearity: in Corollary I, we show f-a f-I spike trains encode temporal information that no finite LIF ensemble can reproduce, under arbitrary Boolean combinations.
7

---

Published as a conference paper at ICLR 2026
4 EXPERIMENTS
In this section, we evaluate f-SNNs on benchmarks spanning neuromorphic event-driven vision and graph domains. Additional experiments on static datasets, including CIFAR10, CIFAR100, and ImageNet, are detailed in Section D.2.3. Across metrics, f-SNNs consistently outperform conventional SNNs. In particular, fractional adaptations of established SNN architectures within the f-SNN framework achieve higher accuracy, comparable energy efficiency, and improved robustness to noise, supporting f-SNNs as an effective extension of traditional SNNs. Importantly our primary aim is not to achieve state-of-the ohtt-oo-the-rtt (SOTA) results, but demonstrate that the generalized f f-SNN framework can improve existing integer-ordes SNNs. To our knowledge, SOTA performance on very large datasets typically requires substantial computational resources, even within the energy-efficient SNN community. We focus on fair comparisons by replacing the integer-order IF/LIF modules (9) in traditional SNNs with the f-IF/f-LIF modules (13) from our f-SNN framework.
4.1 NEUROMORPHIC DATA CLASSIFICATION TASKS
Neuromorphic data event-driven and exhibit strong spatiotemporal correlations. SNNs, with their natural adaptability to spatiotemporal data (e.g., dynamic event processing and sparse coding), neuro-
efficiently model these correlations. Therefore, conducted series of experiments morphic datasets. Our experiments primarily focus on the following key evaluation aspects: (1)
Classification performance, and (2) Robustness of the proposed f-SNN model. More ablation studies and experimental details will be presented in the Section D.
Table 1: Neuromorphic data classification results in terms of classification accuracy (%) on the multiple datasets. The best results are boldfaced, while the runner-ups are underlined.
Datasets/Configs
Architecture
Timesteps
LIF
LIF
f-LIF
(SpikingJelly)
(snnTorch)
(fSNNN
N-MNIST
CNN-based
16
0.9927
0.9908
0.9948
DVS-Lip
CNN-based
16
0.4241
0.3271
0.4342
CNN-based
16
0.9340
0.8899
0.9480
DVS128Gesture
Transformer-based
16
0.9514
0.8715
0.9583
CNN-based
16
0.6682
0.6521
0.7026
N-Caltech101
Transformer-based
16
0.7263
0.6567
0.7627
CNN-based
0.4610
0.4626
0.4766
HarDVS
Transformer-based
8
0,4520
0.4614
0.4723
Dataset & Baselines. We conduct comprehensive evaluations of the Pooosed framework on neuromorphic datasets including N-MNIST (Orchard et al., 2015), DVSI DVS128Gesture (Amir et al., 2017), N-Caltech (Orchard et al., 2015), DVS-Lip (Tan et al., 2022), and the large-scale dataset HarDVS (Wang et al., 2024). The dataset details and experiment setting details are provided in Appendix Section D.
Experimental Setup. For the N-MNIST dataset, set the batch size 512, the number of time steps 7 to 16, and train for 100 epochs using the Adam optimizer. For other neuromorphic datasets, we follow the standard preprocessing pipeline the SpikingJelly framework convert event data into frame representations. For time step configuration. DVS128Gesture, N-Caltech101 and DVS-Lip set 16 time steps, while HarDVS is set time steps due the large data size. N-Caltech 101 is split into training and test sets with an 8:2 ratio. These datasets use batch size of 16, with input dimensions uniformly adjusted 128x128 pixels. We train for 200 epochs using the Adam optimizer.
Classification Performance. f-NN, we have a hyperparameter a indicating the fractional order, which gives the model an additional degree of freedom to capture richer temporal patterns. During experiments, the optimal a is obtained via hyperparameter tuning. The experimental results on neuromorphic datasets are shown Table We conduct comprehensive comparisons between f-SNN and baseline models. The experimental results demonstrate that under the same network configurations, regardless of whether CNN or Transformer architectures are employed, f-SNN significantly and consistently outperforms baseline networks implemented based SpikingJelly and snnTorch frameworks. These results validate the effectiveness and superiority of the proposed S-SNN method in terms of classification performance. This is because f-NNN captures long-term
8

---

Published as a conference paper at ICLR 2026
dependencies in membrane potential via fractional dynamics, enabling richer temporal patterns than traditional models.
Robustness Analysis. We further validate the ro- bustness advantages of f-SNN. We comprehen- sively test the model's stability from five dimen- sions: noise injection, occlude block, temporal truncate, temporal jitter, and discard frame. De- tailed experimental settings are provided in Sec- tion D.1.3, The experimental results are shown in Fig. 4, where f-SNN significantly outper- forms baseline methods across all five robust- ness testing dimensions. Particularly under high- intensity noise injection and occlude block in- terference conditions, our method demonstrates exceptional anti-interference capability and sta- bility. To more intuitively validate viewpoint, we visualize the shallow feature maps with oc- clude blocks, with results shown in Fig. 5. Our f-SNN model can better capture object features under occlusion conditions compared to the other two models. This advantage is primarily attributed the inherent characteristics of the f-LIF neuron module, which can generate more stable and reliable spike patterns, thereby effectively enhancing the noise suppression capability and robustness performance the entire network. We refer the readers to the discussions in Section 3.1. Evaluation details are provided in the appendix.
No occlusion
LIF (SJ)
LIF (suntorch)
ALIF (FSNN)
Figure 5: Feature map visualizations of LIF and f-LIF in occluded block scenarios.
4.2 GRAPH LEARNING TASKS.
For graph learning tasks, our experiments focus on the following key aspects of evaluation: (1) Node Classification performance; (2) Energy Efficiency; and (3) the Robustness of the proposed framework.
Dataset Baselines. We conduct experiments on two mainstream GNN methods: SGCN (Zhu al., 2022), and DRSGNN (Zhao et al., 2024), using several commonly used graph learning datasets. Specifically, Node classification performed with SGCN and DRSGŇN on Cora (MeCallum et al., 2000), Citeseer (Sen et al., 2008), Pubmed (Wang al., 2019), Photo, Computers, and ogbn-arxiv (Hu ct al., 2020). To ensure fairness, we only replace the integer-order neuron modules the baseline models with proposed modules, i.e., the LIF neuron (9) in SGCN and DRSGNN changed to our f-LIF iterations (13). This ensures our fractional adaptations have the same trainable parameters as the baselines; only the charging phases (9) and (13) differ. Dataset details are provided in Section D.
Experiment Setup. For node classification tasks based on SGCN and DRSGNN, we use Poisson spike encoding. The number of timesteps N is 100, and the batch size 32. Datasets are split into
Nošse injoction
Performance Comparioo under Noise Levels Performance Comparison under Occlade Bluck Ratios
1.8
Discard
Performunce
Performance
0.8
Occlude
Temporal jilter
LB toaiii
0.6
110 (q
Temporal truncate
Noise Levels
Occlude Block
Figure 4: Robustness comparison between the proposed f-SNN and two integer-order baselines (LIF SpikingJelly and LIF in snnTorch). Left: Radar chart aggregating five corruption types (larger is better): Gaussian noise injection, center occlude block, temporal truncate, temporal jitter, and discard frame. Middle: Performance vs. noise level (x-axis: Gaussian noise std). Right: Performance vs. occlusion ratio (x-axis: area ratio of the center block). f-LIF (f-SNN) shows consistently higher performance and slower degradation under all corruption types.

---

Published as a conference paper at ICLR 2026
training/validaaiionness with ratios 0.7/0.2/0.1. For DRSGNN experiments, the ceaational-enoddin dimension is 32, using Laplacian (LSPE) (Dwivedi et al., 2023) or random-walk (RWPE) (Dwivedi independently 20 times; we report the mean and
et al., 2021) encodings. All experiments standard deviation. Other experimental details are included in Section D.
Table 2:
Node classification results in terms of classification accuracy (%) on multiple datasets. The best results are boldfaced, while the runner-ups are underlined. Standard deviations are provided as subscripts. The choice of J-SNNs' parameter a will be shown in Table 4.
Methods
Cora
Citeseer
Pubmed
Photo
Computers
ogbn-arxiv
SGCN (SJ)
81.81
01.83
86.7940.32
07.72
70.8640.24
50.26
SGCN (smnTorch)
83.12
20.6
59.82.+1.07
83.3420.80
40.87
21.55
SGCN (f-SNN)
08.08
73.80 61
87.17
92.49 =0.82
89.12+0.21
51.10+0.14
DRSGNN (SJ)
83.30
72.72:10.24
87.1340.38
10.31
76.55
50.1340.14
DRSGNN (ornn)
80,98
68.00
68.00.2...00
59.56 1.05
82.28
76.78
28.46
DRSGNN (f-SNN)
20.61
:5.11:00.45
0..2
91.93_0.20
88.77+0.20
53.13g0.18
Node Classification Performance. The experimental results based on SGCN and DRSGNN shown in Table 2. Our fractional extension of SGCN and DRSGNN outperforms the original versions implemented with traditional integer-order SNN toolboxes (SpikingJelly or snnTorch) in terms of accuracy. These results highlight the clear advantage of our method in improving model accuracy.
Energy Consumption Analysis. Following (Yao et al., 2023a; 2024). we compare the energy consumption of f-SNN and the integer-order method (SpikingJelly). Fig. 6a shows that f-SNN achieves higher accuracy and significantly lower energy consumption across datasets, demonstrating its superior energy efficiency. Details will be discussed in the Section D.3.
Energy Consumption Comparison Across Dutasets
Energy Consumption (nJ)
scca plang
glone
ferformance Comparison under Gaussian Noise
1.00
30
0.75
Accuracy
0.50
20
Computers
Photo
Chunevr
Cure
Pahned
srat
0.00
Datasets
0.00 0.05 0.10
0.20
0.30
0.40
0.50
Noise Stundard Deviation
(a) Comparison of energy consumption between integer- order SpikingJelly and snnTorch baselines and our f- SNN framework.
(b) Robustness test for SGCN.
Figure 6: Energy consumption and robustness evaluation. Best zoomed on screen.
Robustness Test. We further validate the robustness advantage /-SNN. Specifically, we randomly add Gaussian noise (Hall, 1994) of varying intensities the spike signals input to the network to evaluate the robustness of spiking graph neural networks under different noise conditions. The experimental results are shown Fig. 6b.
CONCLUSION
In this work, we introduced new f-SNN framework, which extends traditional SNNs by replac- ing first-order ODEs with fractional-order ODEs to capture the non-Markovian characteristics and long-term dependencies observed in biological neurons. Our experiments demonstrate that f-NNNs consistently outperform integer-order SNNs across neuromorphic vision and graph benchmarks, achieving higher accuracy, comparable energy efficiency, and improved noise robustness. The accom- panying open-source toolbox facilitates adoption of the framework across diverse architectures and applications. These results establish f-NNNs as a promising extension of traditional SNNs, of- fering a mathematically rigorous and biologically plausible approach to enhancing neuromorphic computing capabilities.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China under Grants 62225207, 62436008, 62576326, and 12301491. To improve the readability, parts of this paper have been grammatically revised using ChatGPT OpenAl (2022).
REFERENCES
Larry F Abbott. Lapieque's introduction of the integrate-am -nnggrate-nd--aae fire model neuron (1907). Brain research bulletin, (003-6)3303-1 304, 1999.
Arnon Amir, Brian Taba, David Berg, Timothy Melano, Jeffrey McKinstry, Carmelo Di Nolfo, Tapan Nayak, Alexander Andreopoulos, Guillaume Garreau, Marcela Mendoza, al. A low power, fully event-based gesture recognition system. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 7243-7252, 2017.
Justin M Baker, Qingsong Wang, Martin Berzins, Thomas Strohmer, and Bao Wang. Monotone operator theory-inspired message passing for learning long-range interaction on graphs. In International Conference Artificial Intelligence and Statistics, pp. 2233-2241. PMLR, 2024.
Dumitru Baleanu, Kai Diethelm, Enrico Scalas, and Juan J Trujillo. Fractional calculus: models and numerical methods, volume 3. World Scientific, 2012.
Guillaume Bellec, Darjan Salaj, Anand Subramoney, Robert Legenstein, and Wolfgang Maass. Long short-term memory and learning-tor -aannng-tt---ann in networks of spiking neurons. Advances in neural information processing systems, 31, 2018.
Jan Benda. Neural adaptation. Current Biology, 310):R110-RI16, 2021.
Sander M Bohte, Joost N Kok, and Han La Poutre. Error-backpropagation in temporally encoded networks of spiking neurons. Neurocomputiig, 48(1-4): 2002.
Olaf Booij and Hieu tat Nguyen. gradient descent rule for spiking neurons emitting multiple spikes. Information Processing Letters, 95(6):552-55 558-888 2005.
Alan M. Cohen. Inversion Formulae and Practical Results, pp. 23-44. Springer US, Boston,
MA, 2007. ISBN 978-0-387-68855-8, doi: 10.110079988-0338---8555-------3 __ URL https: -87----------8
Allan D Coop. Hugo Cornelis, and Fidel Santamaria. Dendritic excitability modulates dendritic information processing purkinje cell model. Frontiers computational neuroscience, 4:827, 2010.
Javier Cuadrado, Ulysse Rançon, Benoit Cottereau, Francisco Barranco, and Timothée Masquelier. Optical flow estimation from event-based cameras and spiking neural networks. Frontiers in Neuroscience, 17:1 17:1160034, 2023.
Weihua Deng. Short memory principle and a medictor--coreette approach for fractional differential equations. J. Comput. Appl. Math., 206(1):174-188, 2007.
Yabin Deng, Bijing Liu, Zenan Huang, Xiaojie Liu, Shan He, Qiuhong Li, and Donghui Guo. Fractional spiking neuron: Fractional leaky integrate-and-fig circuit described with dendritic fractal model. IEEE Transactions on Biomedical Circuits and Systems, 16(6):1375-1366 2022.
P. Dhar. The carbon impact of artificial intelligence. Nature Machine Intelligence, 2(8):423-425 Aug 2020.
Kai Diethelm. The analysis of fractional differential equations: an application-orienned exposition using differential operators of Caputo type, volume 2004. Lect. Notes Math., 2010.
Kai Diethelm, Neville Ford, and Alan Freed. Detailed error analysis for a fractional adams method. Numer. Algorithms, 36:31-52, 2004.
11

---

Published as a conference paper at ICLR 2026
Vijay Prakash Dwivedi, Anh Tuan Luu, Thomas Laurent, Yoshua Bengio, and Xavier Bresson. Graph neural networks with learnable structural and positional representations. arXiv preprint arXiv:2110.07875. 2021.
Vijay Prakash Dwivedi, Chaitanya Joshi, Anh Tuan Luu, Thomas Laurent, Yoshua Bengio, and Xavier Bresson. Benchmarking graph neural networks. Journal of Machine Learning Research, 24 (43):1-48) 2023.
Bard Ermentrout. Type membranes, phase resetting curves, and synchrony. Neural computation, 8 (5):979-1001, 1996.
Jason K Eshraghian, Max Ward, Emre Neftci, Xinxin Wang, Gregor Lenz, Girish Dwivedi, Mo- hammed Bennamoun, Doo Seok Jeong, and Wei D Lu. Training spiking neural networks using lessons from deep learning. Proceedings of the IEEE, 1110))1016-1054, 2023a.
Jason K Eshraghian, Max Ward, Emre O Nefici, Xinxin Wang. Gregor Lenz, Girish Dwivedi, Mohammed Bennamoun, Doo Seok Jeong, and Wei D Lu. Training spiking neural networks using lessons from deep learning. Proceedings of the IEEE, (10))1006-1005 2023b.
Wei Fang, Zhaofei Yu, Yanqi Chen, Tiejun Huang, Timothée Masquelier, and Yonghong Tian, Deep residual learning in spiking neural networks. Advances in Neural Information Processing Systems, 34:21056-21069, 2021.
Wei Fang, Yanqi Chen, Jianhao Ding, Zhaofei Yu, Timothée Masquelier, Ding Chen, Liwei Huang, Huihui Zhou, Guoqi Li, and Yonghong Tian. Spikingjelly: An apen-source machine learning infrastructure platform for spike-based intelligence. Science Advances, 9(40) 1480, 2023a.
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-term dependencies. Advances in Neural Information Processing Systems, 36:53674-53687, 2023b.
Nicolas Fourcaud- Trocmé, David Hansel, Carl Van Vreeswijk, and Nicolas Brunel. How spike gener- ation mechanisms determine the neuronal response to fluctuating inputs. Journal of neuroscience, 23(37):1 1640, 2003.
Guillermo Gallego, Tobi Delbrück, Garrick Orchard, Chiara Bartolozzi, Brian Taba, Andrea Censi, Stefan Leutenegger, Andrew Davison, Jörg Conradt, Kostas Daniilidis, et al. Event-based vision: A survey. IEEE Trans. Puttern Anal. Mach. Intell., 44(1):154-180, 2020.
Samanwoy Ghosh-Dastidas and Hojjat Adeli. Spiking neural networks. International journal of neural systems, 19(04):295-300 308, 2009.
Gail Gilboa, Ronen Chen, and Naama Brenner. History-dependent multiple-times time-scale dynamics in single-neuron model. Journal of Neuroscience, 25(28):6479- 6489, 2005.
Yufei Guo, Yuanpei Chen, Xiaode Liu, Weihang Peng, Yuhan Zhang, Xuhui Huang, and Zhe Ma. Ternary spike: Learning ternary spikes for spiking neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 12244-12252, 2024.
Natabara Máté Gyöngyössy, Gábor Eros, and János Botzheim. Exploring the effects of caputo fractional derivative in spiking neural network training. Electronics, 11(14):2114, 2022.
Go Eun Ha and Eunji Cheong. Spike frequency adaptation in neurons of the central nervous system. Experimental neurobiology, 26(4):179, 2017.
Michael JW Hall. Gaussian noise and quantum-optical communication. Physical Review A, 50(4): 3295, 1994.
Kaiming He. Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proc. Conf. Comput. Vision Pattern Recognition, 2016.
Alan L Hodgkin and Andrew Huxley. quantitative description of membrane current and its application to conduction and excitation in nerve. The Journal of physiology. 117(4):500, 1952.
12

---

Published as a conference paper at ICLR 2026
Weihua Hu, Matthias Fey, Marinka Zitnik, Yuxiao Dong, Hongyu Ren, Bowen Liu, Michele Catasta, and Jure Leskovec. Open graph benchmark: Datasets for machine learning on graphs. 00Xi::2005.00687 2020.
Yulong Huang, Xiaopeng Lin, Hongwei Ren, Haotian Fu, Yue Zhou, Zunchang Liu, Biao Pan, and Bojun Cheng. Clif: Complementary leaky integrate-and-fire neuron for spiking neural networks. arXiv preprint arXiv:2402.04663, 2024.
Murat Isik, Karn Tiwari, Muhammed Burak Eryilmaz, and Can Dikmen. Accelerating sensor fusion in neuromorphic computing: A case study on loihi-2. In 2024 IEEE High Performance Extreme Computing Conference (HPEC), pp. 1-7. IEEE, 2024.
Qiyu Kang, Kai Zhao, Qinxu Ding, Feng Ji, Xuhao Li, Wenfei Liang, Yang Song, and Wee Peng Tay. Unleashing the potential of fractional calculus in graph neural networks with FROND. In Proc. International Conference on Learning Representations, 2024a.
Qiyu Kang, Kai Zhao, Yang Song, Yihang Xie, Yanan Zhao, Sijie Wang, Rui She, and Wee Peng Tay. Coupling graph neural networks with fractional order continuous dynamies: A robustness study. In Proc. AAAI Conference on Artificial Intelligence, Vancouver, Canada, Feb. 2024b.
Qiyu Kang, Kai Zhao, Yang Song. Yihang Xie, Yanan Zhao, Sijie Wang, Rui She, and Wee Peng Tay. Coupling graph neural networks with fractional order continuous dynamics: A robustness study. In Proc. AAAI Conference on Artificial Intelligence, Vancouver, Canada, Feb. 2024c.
Shujaat Khan, Jawwad Ahmad, Imran Naseem, and Muhammad Moinuddin. A novel fractional maddient-based based learning algorithm for recurrent neural networks. Circuits, Systems, and Signal Processing, 37(2):593 (222),293361, 2018.
gradient-
Christoph Kirch and Leonardo L. Gollo. Spatially resolved dendritic integration: Towards functional classification of neurons. PeerJ, 8:e10250, 2020.
Christof Koch, Moshe Rapp, and Idan Segev. A brief history of time (constants). Cerebral cortex, 6 (2):93-101, 1996.
Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, 2009.
Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton. Imagenet classification with deep convolu- tional neural networks. Advances in neural information processing systems, 25, 2012.
Giancarlo La Camera, Alexander Rauch, David Thurbon, Hans-R Luscher, Walter Senn, and Stefano Fusi. Multiple time scales of temporal response in pyramidal and fast spiking cortical neurons. Journal of neurophysiology, 96(6))3448-3444 3464, 2006.
Yann LeCun, Bernhard Boser, John Denker, Donnie Henderson, Richard Howard, Wayne Hubbard, and L Lawrence Jackel. Handwritten digit recognition with a back proppppopagntion network. Advances in neural information processing systems, 2, 1989.
Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. nature, 521(7553):436 444, 2015.
Holden T Lee and Daniel M Monahan. Exploring fractional leaky integrate-and-fium neurons in spiking neural networks. Artem Arefev, pp. 138.
Jun Haeng Lee, Tobi Delbruck, and Michael Pfeiffer. Training deep spiking neural networks using backpropagation. Frontiers in neuroscience, 10:508, 2016.
Xinhao Luo, Man Yao, Yuhong Chou, Bo Xu, and Guoqi Li. Integer- valued training and spike-driven inference spiking neural network for hegf-peeformance and energy-efficient object detection. In European Conference on Computer Vision, pp. 253-272. Springer, 2024.
Lei Ma, Fanhai Zeng. Ling Guo, George Em Karniadakis, al. Bi-orthogonal fpinn: A physics- informed neural network method for solving dime-dependent stochastic fractional pdes. arXiv preprint arXiv:2303. 10913, 2023.
13

---

Published as a conference paper at ICLR 2026
Wolfgang Maass. Networks of spiking neurons: the third generation of neural network models. Neural networks, 10(9):1659-167)) 1997.
Michael Mathieu, Mikael Henaff, and Yann LeCun. Fast training of convolutional networks through ffts. arXiv preprint arXiv:1112.5885/ 2013.
Andrew Kachites McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. Automating the construction of internet portals with machine learning. Information Retrieval, 3: 3:127-163, 2000,
Kenneth D Miller and Todd W Troyer. Neural noise can explain expansive, power-law nonlincarities in neural response functions. Journal of neurophysiology, 87(2):653-65) 2002.
Emre Neftci, Hesham Mostafa, and Friedemann Zenke. Surrogate gradient learning spiking neural networks: Bringing the power of gradient-basee optimization to spiking neural networks.
IEEE Signal Processing Magazine, 36(6):5 2019.
Gabriel Nobis, Maximilian Springenberg, Marco Aversa, Michael Detzel, Rembert Daems, Roderick Murray-Smith Shinichi Nakajima, Sebastian Lapuschkin, Stefano Ermon, Tolga Birdal, al. Generative fractional diffusion models. Advances in Neural Information Processing Systems, 37: 13469-25509, 2024.
OpenAI. Chatgpt-4, 2022. Available at: https: / www.wwwww..............ww.w..w.w com (Accessed: 10 April 2024).
Garrick Orchard, Ajinkya Jayawant, Gregory Cohen, and Nitish Thakor. Converting static image datasets to spiking neuromorphic datasets using saccades. Frontiers in neuroscience, 9:437, 2015.
Guofei Pang, Lu Lu, and George Em Kariiadakis. fpinns: Fractional physics-informed neural networks. SIAM Sci. Comput., 41(4):2263--2266, 2019.
Jing Pei, Lei Deng, Sen Song, Mingguo Zhao, Youhui Zhang, Shuang Wu, Guanrui Wang, Zhe Zou, Zhenzhi Wu, Wei He, et al. Towards artificial general intelligence with hybrid tianjic chip architecture. Nature, 572(7767):00-1111 2019.
Igor Podlubny. Fractional Differential Equations. Academic Press, 1999.
Nitin Rathi, Gopalakrishnan Srinivasan, Priyadarshini Panda, and Kaushik Roy. Enabling deep spiking neural networks with hybrid conversion and spike timing dependent backpropagation. In International Conference on Learning Representations, pp. 2019.
Henri Rebecq, René Ranftl, Vladlen Koltun, and Davide Scaramuzza. Events-to-video: Bringing modern computer vision to event cameras. In CVPR, pp. 3857-3866, 2019.
Jaldert Rombouts and Sander Bohte, Fractionally predictive spiking neurons. Advances in Neural Information Processing Systems, 23, 2010.
Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards baike- machine intelligence with neuromorphic computing. Nature, 575(7784):607-617 2019.
Jocelyn Sabatier, Patrick Lanusse, Pierre Melchior, and Alain Oustaloup. Fractional order differ- entiation and robust control design. Intelligent systems, control and automation: science and engineering, 77:13-18, 2015.
Prithviraj Sen, Galilco Namata, Mustafa Bilgic, Lise Getoor, Brian Galligher, and Tina Eliassi- -Rad Collective classification in network data. Al magazine, 29(3):93-93, 2008.
Yeonjong Shin, Jérôme Darbon, and George Em Kamiadakis. Accelerating gradient descent and adam via fractional gradients. Neural Networks, 161:185-201, 2023.
WJ Spain, PC Schwindt, and WE Crill. Two transient potassium currents in layer pyramidal neurones from cat sensorimotor cortex. The Journal of physiology, 434(1):591-60) 1991.
Richard Stein. A thcoretical analysis neuronal variability. Biophysical 5(2):173-194 1965.
Richard Stein. Some models of neuronal variability. Biophysical journal,, 7(1):37-68, 1967.
14

---

Published as a conference paper at ICLR 2026
Li Sun, Zhenhao Huang, Qiqi Wan, Hao Peng, and Philip S Yu. Spiking graph neural network on riemannian manifolds. arXiv preprint vrXiv:2410.199444 2024.
Tayebeh Taheri, Alireza Afzal Aghaei, and Kourosh Parand. Accelerating fractional pinns using operational matrices of derivative. arXiv preprint urXiv:2401.14088,, 2024.
Ganchao Tan, Yang Wang, Han Han, Yang Cao, Feng Wu, and Zheng-Jun Zha. Multi-grained spatio-temporal features perceived network for event-based lip-reading. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.
2022.
Haotian Tao, Ziyan Zhang, Bo Jiang, and Bin Luo. Learning efficient linear graph transformer via
graph-attention distillation. Machine Intelligence Research, 22(6): 1138::111--10 118-118811111 1152, 2025.
Vasily E Tarasov. Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer Science & Business Media, 2011.
Corinne Teeter, Ramakrishnan Iyer, Vilas Menon, Nathan Gouwens, David Feng, Jim Berg, Aaron Szafer, Nicholas Cain, Hongkui Zeng, Michael Hawrylyez, et Generalized leaky integrate-and- fire models classify multiple neuron types. Nature communications, 9(1):709, 2018.
Wondimu Teka, Toma M Marinov, and Fidel Santamaria. Neuronal spike timing adaptation described with a fractional leaky integrate-and- model. PLoS computational biology, 10(3)ee100352,, 2014.
Nachum Ulanovsky, Liora Las, Dina Farkas, and Israel Nelken. Multiple time scales of adaptation in auditory cortex neurons. Journal of Neuroscience, 24(46):10440-104) 2004.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones. Aidan Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017
Xiao Wang, Houye Ji, Chuan Shi, Bai Wang, Yanfang Ye, Peng Cui, and Philip S Yu. Heterogeneous graph attention network. In The world wide web conference, pp. 2022-2032, 2019.
Xiao Wang, Zongzhen Wu, Bo Jiang, Zhimin Bao. Lin Zhu, Guoqi Li, Yaowei Wang, and Yonghong Tian. Hardvs: Revisiting human activity recognition with dynamic vision sensors. In Proceedings of the AAAI conference on artificial intelligence, volume 38, pp. 5615-5623, 2024.
Xiao Wang, Yao Rong, Shiao Wang, Yuan Chen, Zhe Wu, Bo Jiang, Yonghong Tian, and Jin Tang. Unleashing the power of cnn and transformer for balanced rgh-event video recognition. Machine Intelligence Research, 22(6): (01:))0031-1477 2025. doi: 10. 10.10077/11133-005- 1555-3.
Yujie Wu, Lei Deng. Guoqi Li. Jun Zhu, and Luping Shi. Spatio-temporal backpropagation for
training high-performance spiking neural networks. Frontiers in neuroscience. 12:331, 2018.
Ninghui Xu, Lihui Wang, Zhiting Yao, and Takayuki Okatani. Mets: Motion-encoderr time-surface for event-based high-speed pose tracking. International Journal of Computer Vision, pp. 1-19, 2025.
Honggang Yang. Jiejie Chen, Ping Jiang, Mengfei Xu, and Haiming Zhao. Fractional gradient descent method for spiking neural networks. In 2023 2nd Conference Fully Actuated System Theory and Applications (CFASTA), pp. 636-641. IEEE, 2023.
Yi Yang, Richard M Voyles, Haiyan H Zhang, and Robert A Nawrocki. Fractional-ordes spike-timing dependent gradient descent for multi-layer spiking neural networks. Neurocomputing, 611:22666, 2025.
Man Yao, Huanhuan Gao, Guangshe Zhao, Dingheng Wang, Yihan Lin, Zhaoxu Yang, and Guoqi Li. Temporal-wist attention spiking neural networks for event streams classification. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 10221-10230, 2021.
Man Yao, Jiakui Hu, Zhaokun Zhou, Li Yuan, Yonghong Tian, Bo Xu, and Guoqi Li. Spike-driven transformer. Advances in neural information processing systems, 66:64043-64058 2023a.
15

---

Published as a conference paper at ICLR 2026
Man Yao, Guangshe Zhao, Hengyu Zhang, Yifan Hu, Lei Deng, Yonghong Tian, Bo Xu, and Guoqi Li. Attention spiking neural networks. IEEE transactions on pattern analysis and machine intelligence, (5(8):9393-9411) 2023b.
Man Yao, Jiakui Hu, Tianxiang Hu, Yifan Xu, Zhaokun Zhou, Yonghong Tian, Bo Xu, and Guoqi Li. Spike-driven transformer v2: Meta spiking neural network architecture inspiring the design of next-generation neuromorphic chips. arXiv preprint arXiv:2404.00 03366, 2024.
Chengting Yu, Zheming Gu, Da Li, Gaoang Wang, Aili Wang, and Erping Li. Stsc-snn: Spatio- temporal synaptic connection with temporal convolution and attention for spiking neural networks. Frontiers in Neuroscience, 16:1077, 2022a.
Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis. Gradient-enhanced physics-informed neural networks for forward and inverse pde problems. Computer Methods in Applied Mechanics and Engineering, 393:1 14823, 2022b.
Jiqing Zhang, Malu Zhang, Yuanchen Wang, Qianhui Liu, Baocai Yin, Haizhou Li, and Xin Yang. Spiking neural networks with adaptive membrane time constant for event-based tracking. IEEE Transactions on Image Processing, 2025.
Shuo Zhang. Lu Liu, Chunhua Wang, Xiaomeng Zhang, and Rong Ma. Multistability and global attractivity for fractional-order spiking neural networks. Applied Mahhmatics and Computation, 508: 126617, 2026.
Han Zhao, Xu Yang, Cheng Deng, and Junchi Yan. Dynamic reactive spiking graph neural network.
In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pp. 16970-16978, 2024.
Shiyun Zhao, Yang Wu, and Yifan Zhang. Degree-aware progressive contrastive learning for graph combinatorial optimization problems. Machine Intelligence Research, 22(6):1153-1166 2025.
Hanle Zheng, Yujie Wu, Lei Deng, Yifan Hu, and Guoqi Li. Going deeper with directly- trained larger spiking neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 35, pp. 11062-11070, 2021.
Zhaokun Zhou, Yuesheng Zhu, Chao He, Yaowei Wang, Shuicheng Yan, Yonghong Tian, and Li Yuan. Spikformer: When spiking neural network meets transformer. arXiv preprint 22XX::2209.15255. 2022.
Zulun Zhu, Jiaying Peng, Jintang Li, Liang Chen, Qi Yu, and Siqiang Luo. Spiking graph convolu- 2022.
tional networks. arXiv preprint
16

---

Published as a conference paper at ICLR 2026
A RELATED WORK
A.1 SPIKING NEURAL NETWORKS
Traditional artificial neural networks (ANNs) have achieved remarkable success across a wide range of tasks (Krizhevsky et al., 2012: LeCun et al., 2015; Vaswani et al., 2017). However, these models differ significantly from biological neural networks, and their computational requirements far exceed those of the human brain (Dhar, 2020). This discrepancy has motivated the development of Spiking Neural Networks (SNNs) (Maass, 1997; Ghosh-Dastidar & Adeli, 2009; Lee et al., 2016; Wu et al., 2018; Zheng et al., 2021: Zhou et al., 2022), which offer more biologically plausible model by communicating through discrete spikes instead of continuous signals. Their event-driven computation enables significant energy savings, particularly 2019; Pei ct al., 2019). Furthermore, SNNs treat time as an intrinsic component, making them well-suited for applications in time-series prediction and real-time interactive systems (Yao et al., 2023b; Luo et al., 2024; Yao et al., 2021). Early biophysical models such as the Bodgkin-Huxiler model (Hodgkin & Huxley, 1952) offer an accurate description of action-potential generation but are computationally expensive, particularly for large-scale learning tasks. Consequently, simplified neuron models, such as the Integrate-and-FFire (IF) and Leaky Integrate-and-Fire (LIF) models, have become widely adopted in SNN research (Abbott, 1999; Stein, 1965; 1967). Beyond the basic IF/LIF models, variety of extensions have been proposed address different modeling challenges. Adaptive Leaky Integrate-and-Fire (ALIF) SNN neuron incorporates neural adaptation mechanisms, such as adaptive thresholds, enhancing the temporal dependency modeling and working memory capacity of SNNs (Bellec et al., 2018; Benda, 2021). The Generalized LIF (GLIF) introduces more physiologically motivated framework, enabling accurate spike detection and unsupervised differentiation of cortical cell types (Teeter al., 2018). 8). The Complementary LIF (CLIF) model further enhances temporal gradient propagation and long- Long-term dependency learning by incorporating complementary membrane potential state (Huang et al., 2024). The Parallel Spiking Neuron (PSN) model eliminates the need for reset mechanisms, facilitating fully connected temporal modeling that allows for time-step parallelism, which accelerates both training and inference (Fang et al., 2023b). Additional neuron models, such as ternary spikes (Guo et al., 2024) and adaptive membrane time constants (Koch al., 1996; Zhang et al., 2025), further extend the capabilities of SNNs. Despite these advances, most existing SNNs discretize only first-order ordinary differential equations (ODEs), which describe dynamics governed by d/dt terms, and assume a Markovian property, where the state at any time depends only on its immediate past (see (9)) (Maass, 1997; Ghosh-Dastidar & Adeli, 2009; Eshraghian et al., 2023b). While this simplification aids tractability, imposes limitations on expressiveness. Neurophysiological evidence indicates that real neurons exhibit long-range temporal correlations (Gilboa et al., 2005), fractal dendritic morphologies (Coop et al., 2010; Kirch & Gollo, 2020), and interactions among multiple active membrane conductances (La Camera et al., 2006; Miller & Troyer, 2002), leading to dynamics that integer-order. Markovian models capture only imperfectly (Ulanovsky et al., 2004; La Camera et al., 2006; Miller & Troyer, 2002; Spain et al., 1991).
neuromorphic hardware (Roy et al.,
Distinction from Prior SNN Families. The models discussed above all belong the integer-order family, where the subthreshold dynamics can be expressed as a first-order ordinary differential equation (ODE) of the form
dU'() = (Un) Lin (1)),
First-order SNN neuron dynamics:
(20)
di
where Dynamic(-) denotes the specific ceabbrane-potentin update rule used by models such as IF, LIF (cf. (10) and (11)), ALIF, GLIF, CLIF, and other related variants. These approaches mainly explore different choices for the function Dynamic(-), all within the same first-order, Markovian framework.
In contrast, our work introduces a fractional-order SNN framework, which is based on
Fractional-order SNN neuron dynamics: (t) Dynamic(U(M)) (u))), 0 1 (21)
where Dt represents a fractional (nonlocal-in-tin time derivative. This formulation generalizes the integer-order case, which is recovered when 1, and incorporates long-term memory in the membrane potential and spike trains through fractional dynamics. In the main text, we instantiate this framework with fractional IF and LIF neurons, but the same approach can naturally extend to
17

---

Published as a conference paper at ICLR 2026
more complex neuron models, such ALIF, GLIF, and CLIF. Our spikeDE toolbox offers modular implementations that facilitate the realization these fractional variants. We believe this contribution significantly advances the field by introducing nonlocal-in- in-time discrete dynamics to SNN modeling.
A.2 FRACTIONAL NEURAL DYNAMICS AND LEARNING
We first review the prior application of fractional calculus into SNNs/ANNs and then position our contribution accordingly.
Fractional biological neuron modelling and shallow fractional Hopfield spiking network. At the neuron level, th modelling (Teka et al., 2014; Deng et al., 2022) of biological neurons explains spike-frequence adaptation in pyramidal neurons (Ha & Cheong, 2017) and yields more reliable spike patterns under noise (Teka et al., 2014). The work (Rombouts & Bohte, 2010) proposes that neuron's spike-train can be interpreted encoding/decoding efficiency and link fractional dynamics to predictive coding. At the network level, related efforts investigate shallow fractional Hopfield-type spiking networks (Zhang al., 2026). These studies primarily focus on dynamical system properties, proving the coexistence of multiple equilibrium points, solution boundedness, and global attractivity rather than learning representations for complex tasks.
fractional derivative of its input signal. They show
Crucially, prior work is restricted to biological modeling, signal-aprroximation or dynamical analysis of fixed-weight, shallow networks, neglecting the learning problem. We bridge this gap by formulating the first g framework for end-to-end training. This advances f f-SNNs from theoretical constructs to trainable computational paradigm compatible with modern deep architectures (e.g., Transformers), strictly generalizing integer-order SNNs.
Distinction:
Fractional deep learning and fractional differential equation neural solvers. In the continuous ANN domain, fractional calculus has been integrated into deep learning frameworks to enhance expressivity. For instance (Kang et al., 2024a;b) leverage fractional calculus to improve graph neural network performance and robustness, while Nobis et al. (2024) utilizes fractional diffusion processes to improve diversity in generative modeling. Separately, in the domain of scientific computing, Inffrme- Neural Networks (PINNs) have been extended to solve fractional partial differential equations (f-PINNs) (Pang et al., 2019). Subsequent developments have focused on scalability, such as gradient-enhanced variants for convergence (Yu et al., 2022b), and optimized training via operator- matrix methods for high-dimensional problems (Ma al., 2023; Taheri al., 2024)
Distinction: Our fur f-NNN model fundaaenttally differs from these approaches. First, unlike f-PINNs,
which serve as function approximators to solve given fractional equation, we embed fractional dynamics inside the neuron model as a computational engine. Second, unlike fractional ANNs that operate on continuous signals, f fSNNS function in the discrete, event-driven domain.
F Fractional-order gradients for training NNs. A complementary line of research applies fractional derivatives to define gradient operators and learning dynamics for training SNNs/ANNs. For example, fractional gradient descent algorithms (Khan et al., 2018; Shin et al., 2023) replace the standard integer-order gradient update with a fractional counterpart. These methods smooth the optimization landscape, enabling faster convergence and better escape from local minima compared to standard stochastic gradient descent (SGD). In the spiking domain, Gyöngyössy et al. (2022); Yang et al. (2025; 2023) have applied fractional gradients for training SNNs.
These approaches use fractional calculus as an optimization tool to adjust the weight update trajectory. whereas our work embeds fractional dynamics within the neurons themselves. This is analogous to the difference between designing a network optimizer (Adam vs. SGD) versus changing the network architecture (CNN vs. Transformer).
Distinction:
A.3 EVENT CAMERA
Event cameras, as novel bio-inspired sensors, capture pixel-level brightness changes through an asyn- chronous triggering mechanism (Gallego et al., 2020). With microsecond-leeee temporal resolution (equivalent to 10,000 fps) and a high dynamic range (140 dB) (Rebecq et al., 2019; Wang et al., 2025), they provide a groundbreaking solution for perception in high-speed dynamic scenes. Unlike traditional frame-based vision sensors, event cameras only record pixels undergoing changes in the
18

---

Published as a conference paper at ICLR 2026
scene, generating sparse event streams. This data structure not only significantly reduces redundant information but also enables robust perception under rapid motion and challenging lighting conditions. In the field of event stream processing, the asynchronous and sparse nature of event data poses chal- lenges for conventional frame-based CNN algorithms. To address these challenges, researchers have proposed various encoding and processing methods tailored the unique characteristics of event data. For instance, (Neftci et al., 2019) introduced a surrogate gradient- method to transform event streams into pulse sequences compatible with neural network processing, (Xu et al., 2025) proposed Time-Surface (METS), which dynamically encodes pixel-level decay rates in time surfaces to capture the spatiotemporal dynamics reflected by events. This approach successfully addresses the challenge of pose tracking in high-speed scenarios using event cameras. Significant progress has also been made in network architecture optimization. (Yu al., 2022a) developed the STSC-SNN model, which introduces synaptic connections with spatiotemporal dependencies, significantly enhancing the ability of spiking neural networks to process temporal information. Fur- thermore, event cameras, with their low latency and high dynamic range, have demonstrated broad application potential in fields such as robotic control, autonomous driving, and object tracking. For example, (Cuadrado al., 2023) proposed 3D convolution- based spatiotemporal feature encoding method, utilizing a hierarchical separable convolution architecture to greatly improve the accuracy and efficiency of optical flow estimation in driving scenarios using event cameras. On the hardware optimization front, researchers have actively developed systems tailored for efficient event stream processing. (Isik et al., 2024) constructed a neuromorphic vision system based on the Intel Loihi chip, achieving significantly lower power consumption compared to traditional GPU solutions, thus providing critical support for the efficient deployment of event cameras.
the Motion-Enceon
B MORE TECHNICAL DETAILS
B. MORE ABOUT FRACTIONAL CALCULUS
In Section 2.1 of the main paper, we presented the (left) Caputo fractional derivative and discussed numerical schemes for solving fractional-order ODEs. Here provide additional background on fractional calculus that underpins our approach. For clarity, we note that throughout the main paper. all references the Caputo fractional derivative specifically denote the left Caputo fractional derivative D
B.1.1 CLASSICAL DERIVATIVES AND INTEGRALS
For scalar function v(),, the ordinary first-order derivative captures its instantaneous rate of change:
dy((t i((
y(t + At) v(t)
lim A/+0
(22)
dt
At
Let denote the integral operator that assigns each function y(t), which assume to be Riemann integrable on the closed interval [o, 77. its antiderivative starting from a:
Jy(t)= S. u(T)dr for E [0,
(23)
When considering positive integers n E N+, we write J" to indicate the n-fold composition of J, where fL - J and J = J Jm-1 for 2. Through repeated integration by parts, one can show that (Diethelm, 2010)[Lemma 1.1.]:
Jnv(() (n 1n /. ''('- (1-7)" for e N
(24)
1.1.2 EXTENDING TO NON-INTEGER ORDERS: FRACTIONAL. INTEGRALS AND DERIVATIVES
Fractional Integrals Operators: Fractional integrals extend classical integration theory by allowing non-integer orders of integration. Among various formulations, the Hiemann-Loouilnn fractional integrals are particularly fundamental (Tarasov, 201 [ppae 4]. For a positive real parameter a e R+, define the left-sided and right-sided Hoamann--Liovillee fractional integral operators, denoted Jü
19

---

Published as a conference paper at ICLR 2026
and Fight as follows:
1
[ (t-7)a-'y(7 (T() dr,
T(a) v(t)
(25)
J V() = T(a) /
1 /b (T- (r-1))-'((7ddn
where I(a) represents the gamma function, which provides continuous extension of the factorial operation to real and complex domains. The key distinction from classical repeated integration lies in the flexibility of the order parameter: while traditional calculus restricts the order in (24) positive integers, the fractional order a in (25) spans the entire positive real line, enabling a continuous spectrum of integration orders.
Fractional Derivative Operators: Parallel to fractional integration, the concept of a fractional derivative extends the operation of differentiation to arbitrary non-integer orders. This allows for more nuanced understanding of rates of change in complex systems.
One common formulation is the Rievann-Liouville fractional derivative. The left-sided (RL D and right-sided (RL D versions are formally defined by first applying fractional integral and then an integer-order derivative (Tarasov, 2011):
dm mm- y(t) lett)yttt= I(m a) dtm Lo y(T)dr (t T)a-m+1
dm
RL y(t) = dt left
(26)
RL Dg-y(t) = (-1)" dm
(-1)m dm (" y(T)
dem right u(t) m(m - a) dt (T 1 t)a-m+1
Here, m is the smallest integer such that m
S m.
Another widely used definition is the Caputo fractional derivative. The left-sided (D") and right- sided (Dg_) Caputo derivatives are distinct from the Dievvile formulation in the order of operations: they involve first taking an integer-order derivative and then applying fractional integral (Tarasov, 2011). They are defined as follows:
Da -mm- d
d y(t) I'm - a) L'
d ((T) dr (t (t-r)a-ma)
4rm
(27)
dm
(-1)Th
dr v(T)
Di_v(!)= == (-1) right dum v()) T(m a)
1)a-m+I
The Caputo formulation offers several advantages: it produces zero when applied constant functions (matching classical derivatives), and accommodates standard initial conditions in differential equations, making it particularly suitable for modeling physical systems. We therefore choose the Caputo formulation.
A fundamental characteristic distinguishing fractional derivatives from their integer-order counterparts is their inherent non-locality. The integral representations (26) and (27) reveal that fractional derivatives incorporate weighted contributions from the function's entire history the interval [a, t] (for left-sided) or [t, b] (for right-sided). This memory effect contrasts sharply with classical derivatives, which depend only on infinitesimal neighborhoods around the evaluation point. The weighting kernel (t - - m1m+1 determines how past states influence the present derivative value, with the fractional order a controlling the decay rate of this historical influence.
This memory memooy-de nature makes fractional derivatives particularly valuable for modeling systems with hereditary properties, long-range interactions, or anomalous diffusion phenomena. In the limiting case where approaches integer value, these fractional operators smoothly transition to their classical counterparts (Diethelm, 2010), establishing fractional calculus as a genuine generalization of traditional calculus. For instance, when Caputo derivatives reduce to the standard n-th order derivative, ensuring theoretical consistency and practical applicability across the entire spectrum of differentiation orders. When dealing with vector- valued functions, the fractional operators act independently on each component, in direct analogy the multivariate extension of ordinary differentiation and integration.
N. both Riemann-Liouville and
20

---

Published as a conference paper at ICLR 2026
B.2 SURROGATE FUNCTIONS IN INSNN
Training SNNs presents a fundamental challenge: the spiking function (Heaviside step function) is non-differentiable, making standard backpropagation impossible. To address this, similar to other works (Wu al., 2018), we employ surrogate gradient methods that replace the undefined derivative of the step function with smooth approximations during the backward pass. In the main paper, we present the threshold-shifted sigmoid function. Our toolbox also implements other commonly used surrogate functions. We present them in this section.
For all surrogate functions, the forward pass computes the standard Heaviside step function:
1, if x if x <<0
H(x)
(28)
0.
The backward pass, however, replaces (x) with a surrogate gradient s(x). Below, we detail each surrogate function implemented our toolbox spi keDE, which can be chosen freely by users.
B.2.1 SIGMOID SURROGATE
The sigmoid surrogate uses the derivative of the scaled sigmoid function:
Sxigmoid (z) a (Kx)) (1 (xx))
(29)
where (xx) = 1-e-z is the sigmoid function and x is a scaling parameter (default: к = 5.0). This surrogate provides smooth gradients centered around the threshold, with the scale parameter controlling the sharpness of the approximation.
B.2.2 ARCTANGENT SURROGATE
The arctangent surrogate employs the derivative of the arctangent function:
Saretan (I) (xx))
(30)
where k is the scale parameter (default: x = 2.0). This function provides a bell-shaped gradient profile with heavier tails compared to the sigmoid surrogate, potentially allowing gradient flow for neurons further from the threshold.
B.2.3 PIECEWISE LINEAR SURROGATE
The piecewise linear surrogate defines a simple triangular approximation:
if 7-
Sineer (x)
2n 0,
(31)
otherwise
where y defines the width the linear region (default: & = 1.0). This surrogate provides constant gradients within fixed window around the threshold, offering computational efficiency the cost of gradient smoothness.
B.2.4 GAUSSIAN SURROGATE
The Gaussian surrogate uses normalized Gaussian function:
1
x²
Sgaussia (x)
exp
(32)
2
20
where a the standard deviation parameter (default: a = 1.0). This surrogate provides the smoothest gradient profile with exponential decay away from the threshold.
21

---

Published as a conference paper at ICLR 2026
Short Memory Window Width
Layer 2
Layer
Timestep
Timestep
Timestop /
Timestop
Time
Timesteps
Figure 7: f-SNN illustration. Within each layer, the current update aggregates all past states of that layer. For efficiency, the short-memory principle approximates the fractional dynamics by retaining only the last M timesteps.
B.2.5 SURROGATE GRADIENT IMPLEMENTATION
In practice, during backpropagation through spiking layer, the gradient of the loss L with respect to the membrane potential is computed as:
OC
ac
s(u)
(33)
Ou
8s
where represents the spike output and s(u) is the chosen surrogate gradient function. The choice of surrogate function and its hyperparameters impacts training dynamics, with sharper surrogates (larger scale parameters) providing more precise threshold behavior but potentially suffering from vanishing gradients.
B.3 MORE ABOUT FRAMEWORK AND ITS VISUALIZATION
In the main paper, we illustrate the distinct information flow characteristies of f-SNN compared to traditional SNNs in Fig. 1. In conventional SNNs, the iterative nature results in skip connections. In contrast, f-SNN utilizes dense connections, which arise from the weighted summation within the ABM predictor, as described in (13). The diagram can be seen in Fig. 7. Within each layer, the current update aggregates all past states that layer. For efficiency, the short-memory principle approximates the fractional dynamics by retaining only the last M timesteps.
The coefficients (a) [a] [m + 1 m²] define causal memory kernel that decays according to power-law, capturing the historical influence of the system. As depicted in Fig. 8, this decay is "heavy-tailed", meaning that although recent states dominate the influence, the contributions from past events remain significant over time, in contrast to the rapid decay for integer-order system.
For large the decay of input influence follows the relationship:
(o) x (m + 1)a m
Expanding this expression, we get:
( ~ m
1 m -1) m-
for large m. This confirms that the kernel exhibits algebraic decay.
22

---

Published as a conference paper at ICLR 2026
Coefficient ch. for different values
0.5
45
02
Figure 8: f-SNN coefficient (a) visuilization. (a) is causal and decays power law, explicitly encoding memory. This decay profile "heavy-tailed." meaning that while recent states exert the strongest influence, distant past events retain non-negligible impact compared to exponential decay.
C THEORETICAL RESULTS AND COMPLEXITY ANALYSIS
C.1 PROOF OF PROPOSITION
Proof. For the first-order case, we multiply both sides of (5) by e//T, yielding
e/7 dU dt + 1/U = R leel
This simplifies to
d (uelit)
-
R
di
Integrating both sides with respect to E, obtain
U(t)et/= /
[ Ieell di RIeel + C,
where C' is the constant of integration. Solving for Ut), we find
U(t) RI + Ce-t/r.
Applying the initial condition U()) Up. we get Ug = RI. + Co C = Up RIc. Therefore, the solution
U(()= = RT. + (Un RT.)e-1T
For the fractional case, we take more general Laplace transform approach. Applying the Laplace transform to both sides of (11), and using the property
C c(O ((U((()) Us) -- -o
we obtain the transformed equation:
S - -0 U(() RI. 1
Rearranging terms and solving for Us), get:
1
RI.
U(s)
8 + -
-0
0
T3
So we have
5a-1Up
Rl
1
U(8)
+ +
g
1
23

---

Published as a conference paper at ICLR 2026
To simplify the second term, observe the partial fraction identity:
7
78
(8" +
8
Substituting this expression, we obtain:
U(s) 10 Uo
5a
Rle
5 (0-1(U Rlc)
RIe
g +
g +
Taking the inverse Laplace transform and using the identity
Ea( Ea(-tt)).
where (-) is the Mittag-Leffler function, arrive the time-domain solution:
U(t)= RIc + (Uo- RIc) Ea
Tu The power-law tail Ea (-t² ~ r(1-a)e- for large follows from (Diethelm, (heorrm 4.3.j.
C.2 PROOF OF THEOREM
In this section, analyze the robustness of our f-SNN framework under input perturbations. We examine two critical aspects: (i) the temporal evolution of membrane potential perturbations, and (ii) the sensitivity of spike timing to input variations. Our analysis reveals that fractional-order dynamics exhibit distinct robustness properties compared to classical integer-order models.
To facilitate the analysis in this section, we choose F-IF neuron dynamics with the continuous formulation (10):
()U(()= = Tin (t).
From Diethelm, 2010[Lemma 6.2.], the above fractional-order ODE can be equivalently written as the following Volterra integral equation:
(tt) Uo T(a:) / [1- (1-u)a-12 Tin (u)du
1
R
(34)
In the following, we consider a constant current input Iin (1) = la perturbed by a small deviation e, yielding Ieeb (t) Ie + e. Without loss of generality, we assume Uo
Membrane Potential Robustness
We denote by Uclean (t) and (t) the membrane potential under the clean input and perturbed input Le + respectively. From (34), we obtain:
Uclean (t) = RI.
RI. =I((a) / (t - u)a -I(aa
[
RI. TI
(35)
+
Similarly, we have:
U perb (t) R(Te +6) r[(a)
(t- (t- = RiTe + t) rI(a
1
The difference between perturbed and unperturbed membrane potentials evolves as:
Re 1
AU (t) Uelean (1) rT(a+
In contrast, for the classical IF model (limit of a - 1), we have:
AU Re
24

---

Published as a conference paper at ICLR 2026
Since 0 < 1, the perturbation growth follows t (sub-linear) for the fractional model versus t (linear) for the model. This sub-linear accumulation demonstrates that fractional-orde dynamics inherently suppress long-term accumulation of perturbation through memory effects encoded in the fractional derivative.
Spike Timing Sensitivity
The first spike time occurs when the membrane potential reaches threshold 8. We emphasize the necessity of robust spike timing, noting that various learning algorithms rely explicitly on these temporal values (Bohte et al., 2002; Booij & tat Nguyen, 2005; Rathi et al., 2019).
For the f-IF system, from (35), we have:
(OTT(a+1) (1 1/a
gelean =R-Re RIc
Under perturbation 1e - Le + the spike time changes to:
0TI((a11)) 1/a
ppeb
(mR(1e-
(R(T ++)
We define the spike time shift magnitude as At, Iloloan &perbl
(
1/a
Ie
1/a
Ar--FF IF
-RRe
1
(36)
RIc
Ie +
For small perturbations e < lc, using Taylor expansion:
Le Ie
1/a
=(1+ =1
-1/a
+
Ie
=1 I
+ 0(2)
Te
The first-order approximation therefore:
(oI + +11)
1/a
A (IF
Rlc
ale
For the classical IF model, following the same procedure, obtain:
OT
A
IF
RI.
T
Examining the dependence on input current lc, we observe that for the fractional model, the sensitivity decays more rapidly with increasing Ic, the dependence is:
A A IF
x6 c.. T (1+1/a)
For the IF model, it decays as:
At!F I.
Since + 1/a > 2 for 0 < a < 1, this shows that the fractional model is more robust in terms of spike timing for high input currents.
These findings demonstrate that fractional-orde dynamics introduce nuanced robustness profile, with distinct advantages in specific operational regimes rather than universal superiority.
C.3 PROOF THEOREM 2
In this section, we analyze the representational capacity of the f-SNN neuron compared to finite banks of standard integer-order SNN neurons. We demonstrate that fractional dynamics induce an infinite-meman structure that cannot be exactly realized by any finite linear combination of integer- order SNN neurons with arbitrary weights. This structure can only be represented as an infinite continuum of integer-order modes.
25

---

Published as a conference paper at ICLR 2026
Let the discrete time index be E Z20. In the main paper, to discretize the f-SNN with Caputo derivative D for presentation. The literature offers various definitions of fractional derivatives. To facilitate the theoretical analysis in this section, instead (GI.) definition (Diethelm, 2010) here. We emphasize that our f-SNN
fractional ABM predictor
utilize the Grünwald-Leennikee toolbox spikeDE supports multiple fractional definitions, including Caputo, Gl., and others. The analysis here does not restrict the implementation.
Definition 2 (Grünwald Letnikov Fractional Derivative). For a function y(t) defined over interval [0, T). is Grünwald-Letiine fractional derivative of order or E (0, 1] is given by (Diethelm, 2010):
Dāvv(1) = lim h-+0 h 1=
1
Σ (-1y)
(G) v((1 -(h).
(37)
The GL weights are defined as:
- a
(38)
Recall the fractional-order ODE from (3): DE f(t, v(t)). Approximating DaLy (fx) by the finite GL. sum gives:
k
yk-j yk- h15 (44 (4 y/k 9k-
1=0
Without loss of generality, for the neurons, we set T = 1, h 1, and R = 1 to simplify the analysis (the constants here just rescale things and can be absorbed into input X). For simplicity, we primarily consider the f-IF neuron to demonstrate that any finite linear combination of integer-order LIF neurons with arbitrary weights cannot exactly realize it. Noting that efa) 1, the discrete-time membrane potential update for the f-IF, based on the GL definition, is given by:
UUP -K.
(a)
(39)
where 1/ is the membrane potential of fIF neuron.
Recall standard discrete-time LIF neuron with leak factor 3 (0, 1) is governed by the recurrence:
U+ (.I((()))
BU + XNs
U0 (.()(()) 0
By unrolling the recurrence, we obtain the solution
U
LIF(3)
8"m--m=
(1 xx)
m==
where the impulse response is the geometric sequence
3th
m 2
We now consider finite bank of integer-order neurons with leak factors (8:) C (0, and readout weights { C IR. The aggregate output is denoted as:
W
ÚK = Σ SLU
LIF(A))
Theorem 3 (Playback of Theorem 2). Let UT-IF denote the trajectory of a f-FF neuron with order
W (0,1). 1). There exist no finite integer W. weights { and leak factors {3)) such that the following holds:
Ú== = Σ S.U =U.f Vk.
LIF(B.)
for general input Xx- The impulse response error of the approximation is O(ka-1), decaying algebraically slowly. The f-IF neuron mathematically equivalent to an aggregate of integer-order LIF neurons if and only if W - co, specifically as an integral over continuum of leak factors.
26

---

Published as a conference paper at ICLR 2026
Remark 6. The result above emphasizes the infinite-dine linntte-dimensionlll nature of the f-IF neuron compared to any ditte-dimessonnll approximation via integer-ordes neurons. Intuitively, the impulse response of standard integer-order neuron decays exponentially (3" = e- Ak,, characterizing "short-memory process. In contrast, the fractional neuron exhibits an impulse response that decays according to power law (ka-1), characterizing "long-memory" process. finite sum of exponentials never exactly match power law tail.
Proof. Recall from (39) that:
Σ j=0
U Xk:
which represents discrete convolution (e(() UT-1F); XA. Applying the Z-transform converts this convolution into the algebraic product:
C())U())=X((>
where C(2) - 20)2- is the Z-transform of the kernel, and U(2) and (() are the Z-
transforms of U IF and XR. respectively.
Using the generalized binomial theorem, we identify the series expansion of the kernel as:
C(2) - - (:) (-
- -1-2 2
5-0
Thus, the transfer function H (2)
(())
c(2) is the reciprocal of the kernel:
[(=)
(=)= 21-2
Since (1 -----a--a has algebraic branch point at for & Z, it is non-rational.
By the linearity the Z-transform, the aggregate transfer function (2) is the weighted sum of the individual transfer functions. Since the impulse response of the i-th LIF neuron is the geometric sequence (. 8.". its Z-transform is the standard geometric series:
H+(2)
8 m
- 8,2
1===
Therefore, the aggregate transfer function is:
W ===)= H.H(()) Σ Ø:
8,2-1
This is strictly rational function of with degree at most W. A discrete-time linear time-invariant (LTI) system has a dimee--aimnnional state-space realization if and only its transfer function is rational. Since (z) (z) is irrational, it implies that the system possesses an anfinte-ddimensinae state space and cannot be realized by any finite W.
The rigorous link between the the transfer functions is established through the asymptotic decay rates of the impulse responses. The impulse response of the finite bank, h finile, the inverse Z-transform of the rational function (2). This yields linear combination of geometric sequences:
Φi
finite Z hi
B
Let 3max max, [8,]. Since stability requires |3:|
< 1, the decay is bounded exponentially:
1hal
Σ
g
(40)
27

---

Published as a conference paper at ICLR 2026
In contrast, the impulse response of the fractional neuron corresponds to the coefficients of the irrational function (1 - 2-1 )=1) By the generalized binomial theorem, for 2
< 1.
(1-23 21-2
(-1)
a
r(k + + *
k
[(a)l(k
k=
k=0
Thus the coefficient of = is
r(k + a) (k + +1
/I K
Using the asymptotic property of the ratio of Gamma functions, this response follows a power-law decay:
I-
ka-1 k
(41)
r(a)
Comparing the exponential decay of the finite bank and the power-law decay of the fractional neuron, we observe that the exponential decay is strictly faster than the algebraic decay. Hence, for any fixed W. there exists time step T such that for all T", the finite approximation becomes negligible
relative to the fractional signal:
h Anite 1/
lim 1-000
O.
This implies that for sufficiently large k. the finite approximation hy becomes negligible relative to t/g Specifically, there exists a time T such that for all h: > T", [hgl 1 Consequently, the pointwise error at the tail is lower-bounded:
hh
f-1
>
2
(a)
Thus, the "heavy-tailed" memory induced by the irrational transfer function cannot be captured by the "light-tailed" exponential memory of any finite rational system.
Although no finite realization exists, show that infinite representation is valid. Use the Beta- function identity
B(m + a. a) / gmta-1 (1
a dt I(m + a) -1) a) M(m + 1)
0
Divide both sides by T(a) - a) B(a, a) to obtain
r(m a))
em ta (1 -1--)-a
dt.
[ma)[(++]]
J0
Bla, 1 -x)
Hence, with
dv, (B) = ga-1 33- d3
B(a, 1 a))
we have
h/-IF
8thh dv (3).
Jo
Therefore, there exists a spectral measure Va supported on [0, 1] such that:
(
Uf
Σ
3th dv
X Xa-1-m
By Fubini's theorem (the sum and integral are both over non-negative indices and the integrand is non-negative), we may exchange the summation and integration:
Uf
Σ
3Th X
dva
['
U LIF(3) dva
where U LLIF( is the state of a LIF neuron with leak factor 3.
This establishes that the fractional neuron state Uf- is the aggregate output of population of LIF neurons U. (IP(()) distributed over the leak factor 3, (0, 1] according to the density v,, In other words, the fractional neuron state is continuous mixture of integer-order LIF neurons.
n
28

---

Published as a conference paper at ICLR 2026
C.4 COROLLARY: SPIKE TRAIN IRREDUCIBILITY
Theorem 2 establishes irreducibility at the membrane potential level, but the functional output of spiking neuron is the spike train. We now clarify the connection between membrane potential dynamics and spike output expressivity.
Clarilication: From Membrane Potential to Spike Output
The spike generation mechanism directly couples membrane potential to output: spike is emitted when UR 2 (threshold). Therefore, the spike train Sp E {0, 13 is deterministic function of the membrane trajectory:
S H (Uk -
where H-) is the Heaviside step function. This coupling implies that differences in membrane potential dynamics propagate differences spike train patterns.
We formalize this connection with the following corollary:
Corollary 1 (Spike Train Irreucibilityy. Let 0 < < 1. For any finite integer W, weights W.W C IR, leak factors {B C (0, and any Boolean function f: {0, 1] {0,1), there exists an input sequence {X& and threshold >> O such that the spike train of the f-IF neuron cannot reproduced by any Boolean combination of spike trains from the W LIF neurons. That is,
Sf * S
for some k.
(42)
This corollary establishes that the expressivity advantage of f-IF neurons extends to the spike train level. While Theorem demonstrates irreducibility in membrane potential dynamics, one might wonder whether this advantage could be "washed out" by the thresholding nonlinearity. The corollary confirms is not: the spike patterns generated by a single f-IF neuron encode temporal information that finite ensemble of LIF neurons reproduce, even when their outputs are combined through arbitrary Boolean logic. This implies that f-SNNs possess fundamentally richer spike-based representations, enabling them communicate long-range temporal dependencies through their spike trains in ways that conventional SNNs cannot.
Remark 7.
Proof. We prove this by construction using an impulse- sillsse---ient---e trieetrr sequence. This construction explicitly demonstrates the role of long-range temporal memory.
Step 1: Input design. Define the input sequence as
A
if = 0, if Fl<k<Tr T
X₁
0
(43)
8 if T.
where A, >> are chosen appropriately, and T is a large delay parameter.
Step 2: Membrane potential at detection time T. At time T. the membrane potential comprises two components: (i) the memory of the initial impulse A, decayed over time steps, and (ii) the immediate response to the test pulse 5.
For f-IF:
UY = A. 11 + 16 f-IF A 16 + 8,
(44)
where Ii-IF ~ 7a-1 (Ta)) exhibits power-law decay.
For the finite LIF ensemble:
Ür = A hr +6,
(45)
W where ho = ΣW 1 T.
Step Asymptotic separation. Define M Σw l and Bmax = max, |8, 1. Then
|hr| S M T
(46)
29

---

Published as a conference paper at ICLR 2026
The ratio of memory contributions satisfies
h2
MS [(a)
T- 0,
(47)
7a-1
since exponential decay dominates power-law decay.
Step 4: Parameter selection. For any fixed finite ensemble (W; [ (Bi)), choose T sufficiently large such that
hTl < 4 11
(48)
Set the remaining parameters as follows:
A 1,
8 1 I-F
0 1 +
I-F
f-I
Step 5: Spike analysis. We analyze the spike behavior at two critical times.
At time T: F-IF,
1T
IF
3 3/
fIF
>
(49)
so the f-IF neuron spikes: ST IF
1.
For each LIF neuron in the ensemble,
jûrl $ hhrl + 6 < 1 + 1 = = < <0,
(50)
so all LIF neurons are silent: ST sLIF(A) for all e (1...., W}.
At times 0 < k < T: The f-IF membrane potential satisfies
Ufa A h/ S A 16 = 1 < 0,
(51)
so f-IF neuron is silent: S IF 0. Similarly, all LIF neurons remain silent during this interval.
Step 6: Establishing a mismatch for any Boolean function. At time T, all LIF neurons are silent, so
[ (SLI)()) ..... ST St (LIF(SW) -(O,...,0)) 0). We consider cases based on the value of f(0, -..,--..0)) 00:
(52)
Case 1: f(O...., 0) = O, then at time T,
Sf = + = (()...,,)
(53)
Case 2: If f(O,................ ..) 0) then at any time 0 < < T; T, all neurons are silent,
=h 1= f
(54)
In both cases, the spike trains differ for some k. completing the proof.
C.5 COMPLEX ANALYSIS
In the iteration of (13) using the ABM predictor (12), each time-step tú we must evaluate the
fractional derivative f (t,, Uj) which is R Tia (t) in f-F and - (t) + R Ji (t) in f-LIF neuron. If there are N T/h steps total, then summing the base cost per evaluation each time step for
all layers with the growing cost (kk) of accumulating k-term histories yields (C O(k)) = O (NC + N2). By leveraging a fast convolution routine (e.g. the FFT-based method of (Mathieu et al., 2013)). the quadratic term can be reduced to O log N), giving an overall forward-pan cost of O + N log N). Since each step also stores its hidden state vector of dimension d, and computing f at one timestamp incurs a peak memory P. the forward memory requirement grows as O(P + Nd), where the Nd term accounts for saving all N evaluations {f (t,, ())); Finally, note that one trim the O(Nd) storage of all past f (ty, y,) values down to ((Md) by invoking the "short-memory" approximation, keeping only the most recent M terms in the iterations. The experimental computational complexity presented in Section D.2.3.
30

---

Published as a conference paper at ICLR 2026
D IMPLEMENTATION DETAILS, DATASET SPECIFICS AND MORE EXPERIMENTS
D.1 EXPERIMENT SETTINGS
D.1.1 GRAPH LEARNING TASKS.
Datasets & Baselines. Our experiments evaluate model performance on diverse collection of graph-structured datasets spanning multiple domains, following standard preprocessing protocols geometric deep learning, with their detailed statistics summarized in Table 3. Specifically, node classification is performed with SGCN and DRSGNN on Cora, Citeseer, Pubmed, Photo. Computers, and ogbn-arxiv. Additionally, we conduct link prediction experiments with MSG-based methods (Sun al., 2024) Computers, Photo, CS, and Physics, with results presented in the following section Section D.2.1.
Table 3: Dataset Statisties of Node Classification Task
Name
# of Nodes
# of Classes # of Features
# of Edges
Cora
2.708
1,433
10,556
Pubmed
19,717
500
88,648
Citeseer
3,327
3,703
9,104
Photo
7,650
8
745
238,162
Computers
13,752
10
767
491,722
OGBN-Arxiv
169,343
40
128
1,166,243
CS
18,333
15
6,805
163,788
Physies
34,493
8,415
495,924
Training& Inference settings. For node classification tasks based on SGCN and DRSGNN, we use Poisson encoding generate spike data following Poisson distribution. The number of timesteps N is set to 100, and the batch size is set to 32. The dataset is divided into training, validation, and test sets in the ratio of 0.7, 0.2 and 0. Additionally, for experiments based DRSGNN, set the dimension of the positional encoding to 32 and adopt the Laplacian eigenvectors (LSPE) (Dwivedi al., 2023) random walk (RWPE) (Dwivedi et al., 2021) method. For link prediction tasks on MSG, we follow the experimental settings described in the original MSG paper. Specifically, we use IF neurons and the Lorentz model, set the dimension of the representation space to 32, and configure the timesteps to [5, 15]. The optimizer used is Adam, with an initial learning rate of 0.001. All experiments are independently run 20 times, with the mean and standard deviation reported.
Table 4: The selection of the parameter a, we report the values that yield the best results on each dataset.
Methods
Cora
Citeseer
Pubmed
Photo
Computers
ogbn-arxiv
SGNN (f-SNN)
0.3
0.3
0.9
1.0
0.8
1.0
DRSGNN (F-SNN)
0.3
0.3
0.8
1.0
0.8
1.0
Selection of Different a Values. Table shows the parameter a we chose in the graph learning tasks.
The corresponding results are shown Table 2.
D.1.2 NEUROMORPIIC DATA CLASSIFICATION TASKS.
Datasets & Baselines. We conduct experiments on five visual classification tasks, including N- MNIST (Orchard et al., 2015), DVSI 1VS128Gesture (Amir et al., 2017), N-Caltech101 (Orchard et al., 2015), DVS-Lip (Tan et al., 2022), and the large-scale dataset HarDVS (Wang et al., 2024). N-MNIST dataset that converts the classic MNIST handwritten digit dataset into neuromorphic event data. It generates event streams by observing moving MNIST digit images on a screen through a Dynamic Vision Sensor (DVS), covering 10 digit categories (0-9). DVS128Gesture is neuromorphic
31

---

Published as a conference paper at ICLR 2026
dataset collected using Dynamic Vision Sensor (DVS) event camera, designed for dynamic gesture recognition tasks. The event camera captures pixel-level brightnsss changes with microsecond temporal resolution. The dataset includes 1,342 samples across 11 gesture categories, such as clockwise rotation, counterclockwise rotation, and left-hand waving. N-Caltech is an event camera-based dataset derived from the traditional static image dataset Caltech101. The original Caltech 101 dataset contains 101 object categories (e.g., animals, vehicles, household items), with each category containing 40 to 800 static images. By simulating translational, rotational, and other movements, event cameras dynamically capture these static images to generate the corresponding neuromorphic data. Similar to DVSI DVS128Gesture, the data from N- X-Caltech101 represented as event streams, which encode pixel-wise brightness changes over time. The classífication task focuses on recognizing object categories in dynamic scenes. DVS-Lip an event camera dataset specifically designed for lip reading, utilizing the high temporal resolution characteristics of event cameras to record fine- aeeegrained changes in lip movements. HarDVS a large-scale human action recognition event dataset containing over 100,000 temporal event stream samples, covering 300 different human activity categories.
To ensure fairness and comparability, we employ two backbone network architectures, CNN and Transformer, on the same datasets, differing only in the choice of spiking neuron modules. The baseline methods use the neuron LIFNode module from the SpikingJelly framework (Fang et al., 2023a) and the snn. leaky module from the snn onnco framework (Eshraghian et al., 2023a), respectively, while our method adopts the f-LIF neuron module (13) defined in A-SN.. The CNN- based-SNN network architecture follows the design of DVSNet in SpikingJelly, while the Transformer- based-SNN uses the structure of Spikformer (Zhou et al., 2022) as the backbone network.
Training& Inference settings. For the N-MNIST dataset, we set the batch size to 512, the number of timesteps T to 16, and use the Adam optimizer train for 100 epochs.
For other neuromorphic datasets, we follow the standard preprocessing pipeline of the Spiking- Jelly framework, converting event data into frame representations. For timesteps configuration. DVS128Gesture, N-Caltech101, and DVS-Lip are set to 16 timesteps, while HarDVS is set to 8 timesteps. N-Caltech101 is split into training and test sets with an 8:2 ratio. All neuromorphic datasets use batch size of 16, with the input size uniformly adjusted to 128x pixels. For CNN-based models, training conducted using the Adam optimizer for 200 epochs, while Transformer-based models trained for 500 epochs.
D.1.3 ROBUSTNESS ANALYSIS SETTING
We validate the robustness advantages of f-SNN in Section 4.1. Specifically, we comprehensively test the model's stability from five dimensions: noise injection, occlude block, temporal truncate, temporal jitter, and discard frame.
Noise Injection: Real-world applications often encounter sensor noise and environmental interfer- ence, which challenges the model's ' stability under noisy conditions. To evaluate this robustness, we randomly add Gaussian noise of different intensities to the input spike sequences.
Real-world scenarios frequently involve occlusion and partial feld-off-iew loss, testing the model's ability handle incomplete visual information. Accordingly. we place square blocks of different sizes at the center of input frames to assess the model's robustness to local information loss.
Occlude Block:
Temporal Truncate:
Practical data collection often results in incomplete temporal information due to various constraints, challenging the model's performance with partial temporal data. Thus, we randomly truncate a portion of temporal data by proportion to evaluate the model's adaptability to incomplete sequences.
Temporal Jitter:
Real systems often suffer from temporal synchronization errors and clock drift, affecting the precise timing of spike events. To simulate these timing uncertainties, we add random temporal offsets to spike events in the time dimension.
Data transmission and processing systems frequently experience packet loss and intermittent data missing, testing the model's tolerance to discontinuous input. Consequently, randomly discard partial frames in the temporal sequence to evaluate the model's 's robustness to data loss.
Discard Frame:
32

---

Published as a conference paper at ICLR 2026
D.2 EXTENDED EXPERIMENTAL RESULTS
D.2.1 LINK PREDICTION.
We have demonstrated the effectiveness of the f-SNN framework on graph node classification. Here, we present additional results for the graph link-prediction task using the graph SNN MSG (Sun et al., 2024). Our fractional adaptation, MSG (f-SNN), consistently outperforms MSG (SJ) across all datasets terms of area under the ROC curve (AUC). Specifically, MSG (S-SNN) achieves the best AUC the Computers, Photo, CS, and Physics datasets, with scores of 94.91 %, 96.80% 96.53% and 96.57%, respectively.
Table 5:
Link prediction results in terms of Area Under Curve (AUC) (%) multiple datasets. The best results are boldfaced.
Methods
Computers
Photos
CS
Physics
MSG (SJ)
24.65
96. 06.75
05.19
93.43
MSG ((-SNN)
94.91
96.80 20.16
10
96.57.0.08
As shown in Table 5, compared the MSG method implemented with the integrator- based approach (SpikingJelly), our f-SNN achieves sigmificant improvements in link prediction tasks. These results underscore effectiveness of
f- -SNN across diverse datasets.
D.2.2 ABLATION STUDIES, CHOICE OF NUMERICAL SCHEMES AND PARAMETERS
In this section, will conduct ablation experiments various hyperparameters in fSNN, including the selection of different order a values, whether to set learnable a, whether to set learnable neuron thresholds, different timestamps T, different network parameters, and different neuron selections. All experiments, unless otherwise specified, are tested on CNN-based SNN and the DVS128Gesture dataset.
Selection of Different a Values: In our /-SNN architecture, different a values can impact experi- mental results. We evaluate the network performance under various value conditions.
0.2
0.4
0.5
0.6
0.8
1.0
Learnable
Acel
0.9336
0.9236
0.9480
0.9193
0.9143
0.9340
0.9362 (Final a: 0.5083)
Table Ablation study on different a values
The experimental results demonstrate that the network does not achieve optimal performance when 1, which further validates the superior performance of f-SN compared to conventional SNN. Notably, setting a as a learnable parameter also yields promising results. Although the accuracy is slightly lower than manual hyperparameter tuning, it still outperforms the case when 1. Moreover, the converged final a value closely approximates our manually fine-tuned result, indicating the effectiveness of the learnable parameter approach.
Different Network Parameters To verify the experimental effectiveness of our f-SNN model on larger parameter models under different timesteps, we tested the performance of the f-SNN model under different parameters and different 7' values. Since the performance on DVS128Gesture has already approached saturation, we conducted ablation experiments on N-Caltech101.
Channel 128 (1.7M)
Channel 256 (4.5M) Channel 512 (13.7M)
LIF(Spiking/elly) LIF(snnTorch)
0.6682
0.7053
0.7108
0.6521
0.6765
0.7423
f-LIF(S-NNN
0.7026
0,7416
0.7684
Table 7: Performance comparison under different network parameters on N-Caltech101
can be seen that with increased parameter count, our f-SNN consistently maintains leading performance, validating the excellent performance of /-SNNN.
33

---

Published as a conference paper at ICLR 2026
D.2.3 STATIC DATASET TESTING
In addition to testing on neuromorphic datasets, also evaluated our method on traditional static datasets, including CIFAR-10, CIFAR-100 (Krizhevsky & Hinton, 2009), and ImageNet (Krizhevsky al., 2012). For the relatively smaller CIFAR-10 and CIEAR-100 datasets, adopt Spiking-ResNet ResNet- 18 (Fang et al., 2021) as the baseline. For ImageNet, we follow the SpikFormer (Zhou et al., 2022) configuration with 29.7M parameters and set both the training and validation image sizes 160x 160 for a fair comparison. Additionally, we include experiments with ImageNet generated using the Beornil spike encoder. The results are shown in Table 8. f-LIF achieves clear gains over both LIF SpikingJelly and LIF snnTorch on all datasets.
Table 8: Comparison of LIF variants across datasets and architectures.
Datasets
Architecture
Timesteps
LIF
LIF (onnhorch)
f-LIF (J-SNN)
CIFAR-10
Spiking- Spiking-ResNet-D Spiking-ResNed ResNet-18 SpikFormer SpikFormer
0,9134
0,9026
0.9215
CIFAR-100
0.6813
0.6445
0.6874
ImageNet
0.6637
0.6584
0.6791
ImageNet (spike encoder)
0.5549
0,5432
0.5738
As shown in the results, our proposed fSNNN demonstrates superior performance on static datasets as well.
Different Neuron Type. To validate the effectiveness of f-LIF neurons, we further compare the performance of different neuron types. In addition to f-LIF neurons, we also f-Iff neurons and traditional IF neurons (SJ) under the same network architecture. The experimental results are shown Table On the 18Se28Gee dataset, the CNN-based-SNN with f-IF neurons achieves an accuracy of 93.833.. 93,83% while the version using traditional IF neurons achieves 92. 70%, representing 1.13 percentage point improvement for f-IF neurons over traditional IF neurons. On the N-Caltech101 dataset, f-IF neurons also demonstrate significant advantages, achieving accuracy of 69.23% compared to 66.59% for traditional IF neurons, representing an improvement of 2.64 percentage points. These results indicate that proposed functionalized neuron design (whether f-LIF f-IF) can bring significant performance improvements compared to traditional spiking neurons. The funetionalized design effectively enhances the learning capability and expressive power of spiking neural networks through more flexible dynamic characteristics.
Table 9: Performance comparison of different neuron types
Dataset
f-IF(--SNN)
(SJ)
Improvement
DVSI 28Gesture
0.9383
0.9270
+1.13%
N- N-Caltech101
0.6923
0.6659
+2.64%
Memory Parameter Analysis
To evaluate the effectiveness of the memory parameter in the f-SNN framework for accelerating training and saving memory, conduct experiments with different memory settings. The experiments are performed with batch size 16 and timesteps 7=16. The results are shown in Table 10. These results demonstrate that the memory parameter in f-SNN framework provides effective mechanism for balancing memory consumption and training speed. Users can adjust this parameter according to their hardware constraints and training requirements achieve optimal performance.
D.2.4 EXTENDED ROBUSTNESS EXPERIMENTS.
To further verify the robustness advantages of FSNN, design and conduct series of experiments for investigation.
Graph Learning Tasks.
Robustness to Feature Masking Ratios in Graph Learning. To evaluate the robustness of the network in graph learning tasks, conduct feature ablation experiments two benchmark datasets:
34

---

Published as a conference paper at ICLR 2026
Table 10: Performance comparison of different memory parameters (Batch=16, 7=16)
Short Memory Size M
Memory (GB)
test-speed(ings/s)
train-speed(ings/s)
Acc
2
14.8
190
32.0
0.9175
16.9
183
29.5
0,9164
17.1
175
29.1
0.9158
8
17.5
163
28.3
0,9130
17.5
156
27.5
0.9158
12
17.8
153
27.3
0,9137
18.3
151
27.0
0.9270
16
19.8
151
26.8
0.9362
Cora and Citeseer. Specifically, we implement random feature masking strategy where proportion of features in the graph feature matrix zeroed according to predefined ratios. This tests the irregular sampling scenarios with partially observed multivariate event series. The modified feature matrices are then fed into the models for prediction accuracy evaluation. As shown in Fig. our proposed f-SNN method demonstrates significantly improved performance compared to the seggrator-methode (SpikingJelly or snn moTocch) baseline method across all tested missing rates. This consistent superiority under varying feature dropout scenarios substantiates the enhanced robustness of our approach against input perturbations.
Performance Compartson under Feature Masking
Performance Comparison under Feature Masking
0.8
Perfrmanee
Performence
0.6
0.5
SLIFI-SNN LiFiSphing)
E1F(Spiking
LiFisan Foech)
03
5.5
61
82
63
54
0.5
6.7
8.8
115
0,1
0.0
02
04
56
0.5
Masking Ratio
Masking Ratio
(a) Accuracy Feature Masking Cora.
(b) Accuracy vs. Feature Masking Citeseer.
Figure 9: Robustness Comparison: Baseline vs. f-SNN in Graph Learning under Feature Dropout Perturbations.
Perform Comparicon Edge Droping
Performance Comparison Edge Droping
0.95
0.85
8.50
Tefformacce
0.50
8.78
8.75
FLIFESNN)
ELIFFESNN)
L(PSpkingg)el)) oly)
11P0ppinggaa)
0.65
LiFisanTiees Trech
8.55
LiFisudodd
8.6
61
03
0.5
87
68
03
6.58
0.0
01
82
04
8/
03
Droping Ratio
Droping Ratio
Accuracy Edge Dropping Photo.
Accuracy Edge Dropping Computers.
Figure 10: Structural Robustness Evaluation: Baseline vs.f-SNN in Graph Learning under Edge Dropping Perturbations.
Structural Robustness Under Edge Dropping Scenarios. To comprehensively evaluate the robust- ness of the network graph-structured data learning. we conduct edge perturbation experiments on implement random edge dropping strategy where a predefined proportion of edges in the graph adjacency matrix are randomly zeroed out according to controlled corruption ratios. The modified adjacency matrices are then utilized for model
two datasets: Photo and Computers. Specifically,
35

---

Published as a conference paper at ICLR 2026
evaluation through standard prediction accuracy metrics. As illustrated in Fig. 10, our proposed f-SNN framework demonstrates superior performance compared to the SpikingJelly or snn baseline across varying edge dropping rates. This consistent advantage under structural perturbations validates the enhanced robustness of our method, particularly in maintaining predictive stability when encountering incomplete graph information.
Neuromorphic Data Classification Tasks. We supplement all robustness test data here and provide specific list of the tests. As shown in Fig. 11 and Tables 11 to 15. To evaluate the robustness of models under corrupted frame conditions, we propose a weighted scoring method. This method is suitable for various frame corruption scenarios, such as frame discarding. noise perturbations, and occlusions, By quantifying model performance under different corruption levels, this method provides a comprehensive assessment of robustness. Specifically, for several corruption levels (e.g., 10%, 20%, 30%, etc.), the model performance is recorded and normalized using its original performance (i.e., performance under no corruption). The normalized performance values are then weighted and summed, and the final weighted score is normalized to range of 0 100 the robustness score. The calculation formula is as follows:
Performance w/: Original Performance, 100,
Final Score,
where:
represents the corruption condition (e.g.. frame discard ratio, noise intensity, etc.);
represents the model;
Wl is the weight assigned to the i-th corruption condition;
Performance,,, and Original Performance, are the model's performance under the i-th corruption condition and the original condition, respectively.
In this study, to simplify the experimental design and ensure fair comparisons across models, we assign equal weights to all corruption conditions, i.e., w; N. where N is the total number of corruption conditions. This choice avoids introducing any bias and ensures that the robustness score calculation remains objective.
This method provides a unified and intuitive metric to quantify the performance degradation of models across various corruption scenarios, offering a standardized basis for robustness evaluation.
Performance Comparias under Temporal Truncrts Leves
Performaace Comparisen under Temperal
Comparioo
1.80
8,50
8.75
8.55
1.90
Porfommnee
8.32
Performance
0.32
4.35
0.94
0.01
60
Temporal Truncate
Temporal Jitter Level
Discard Frame Levels
Figure 11: Robustness comparison between traditional SNN f-NNN framework.
36

---

Published as a conference paper at ICLR 2026
Noise
f-SNN
SJ
snnTorch
0.1
0.9479
0.9236
0.8438
0.2
0.9379
0.9201
0.8299
0.3
0.9236
0.9097
0.7674
0.4
0.9132
0.7326
0.7153
0.5
0.9028
0.5139
0.6667
Score Original
95.96
78.81
79.12
0.9480
0.9340
0.8899
Table 11: Performance comparison under noise conditions
Discard Frame
f-SNN
SJ
snnTorch
0.1
0,9236
0.9137
0.8715
0.2
0,9062
0.9023
0.8403
0.3
0.9062
0.8854
0.8381
0.4
0.8993
0.8646
0.8472
0.5
0.8646
0.8472
0.7967
Score Original
94.93
94.50
94.25
0.9480
0.9340
0.8899
Table 12: Performance comparison under frame discard conditions
Temporal Jitter
f-SNN
snn'Torch
0.1
0.9167
0.9097
0.8576
0.2
0.9132
0.9306
0.8542
0.3
0.9236
0.8681
0.8264
0.4
0.9306
0.8646
0.8299
0.5
0.9062
0.8681
0.8194
0.6
0.8750
0.8472
0.8125
0.7
0.8785
0.8542
0.8299
0.8
0,9028
0.8646
0.7604
0.9
0.8889
0.8368
0.7361
Score Original 0.9480
95.35
93.31
91.47
0.9340
0.8899
Table 13: Performance comparison under temporal jitter conditions
Temporal Truncate
f-SNN
SJ
snn'Torch
0.1
0.9410
0.9271
0.8646
0.2
0.9410
0.9271
0.8611
0.3
0.9271
0.9235
0,8600
0.4
0.9167
0.9018
0.8396
0.5
0.8237
0.8026
0.8125
Score Original
95.98
95.97
95.24
0.9480
0.9340
0.8899
Table 14: Performance comparison under temporal truncate conditions
Occlude Block
S-SNN
SJ
snn'Torch
0.1
0.9340
0.8993
0.8681
0.2
0.9340
0.8576
0.8368
0.3
0.8785
0.7917
0.8264
0.4
0.8368
0,7083
0.7396
0.6840
0.5764
0.6493
Score Original
90.02
82.08
88.10
0.9480
0.9340
0.8899
Table 15: Performance comparison under occlude block conditions.
37

---

Published as a conference paper at ICLR 2026
D.3 ENERGY CONSUMPTION ANALYSIS
Most existing SNN energy analyses primarily account for synaptic operations, while there is no widely adopted methodology for the intrinsic energy of neurons themselves. Therefore, we follow the commonly used practice to estimate the overall SNN energy. In the highlighted energy-analysie part, we use the same methodology as prior work(Yas et al., 2023a; 2024). For neuron-ine inttoniintrinsic costs, we provide the following notation and derivations. The specific algorithm is as follows:
Notation
T: Number of timesteps.
k = AC Ac: Conversion factor from one multiply-accumulac (MAC) operation to equivalent additions (EAc).
Eac: Energy of one equivalent addition; EMAC: Energy of one multiply-accumile operation.
Relation: ÉMAC (K + 1) EAC.
N: Number of SNN convolutional stages.
M: Number of SNN fully connected layers.
L: Number of self-attention (SSA) blocks.
FLOPs((): Floating-ponl operations layer in its dense counterpart.
AMA Energy per caltiply-accumulate operation.
EAc: Energy equivalent addition.
p: Average firing rate.
Following (Yao al., 2023a; 2024), we assume that the data for various operations are implemented using 32-bit floating-poii point arithmetic in 45mm technology, where ÆMAC 4.6p. and EAC = 0.9p..
LIF (discrete time, hard reset; single neuron, T steps) Per-term energy (in units of EAC):
Eupdate.in = T k. EAC
(update: input multiplication),
(55)
Ereak - EAC
(leak: multiplication),
(56)
Eupdate.sum = T 11 EAC
(update summation: add the two terms),
(57)
Eemp T EAC
(threshold comparison),
(58)
Expike T' p- (2x + 1) EAC
(spike: hard reset, arithmetic gating).
(59)
Total energy of single LIF neuron over T steps:
Emeuron ELLI hard
T [222 + 2p) K (2+ p)] EAC-
(60)
II.f-IF (discrete time, hard reset; single neuron, steps) Per-term energy (in units of EAC)):
F Pupdate=teak T log, T (1+1), FAAC (update leak: merged),
(61)
Eemp = T EAG (threshold comparison),
(62)
Expike T p. 11 EAC (spike: hard reset, arithmetic gating).
(63)
Total energy of single /LIFF neuron over T steps:
Ef Ef-LIF hard T [(x 1) T p(2x =1]] EAC
Ensuron
E
(64)
Energy accounting for Spiking-Transforee Transformer (SpikFormer). Based on the updated formula, we further consider more complex Spiking-Transforre setting and compare energy consumption accordingly. The energy of SpikFormer can be written as
ESpikFormer -EMAC x FLOP + EAC
SSNN Coav
(65)
Σ
SOP SNN Conv
Σ
SOP m SNN FC
+
SOP! SSA
m=
38

---

Published as a conference paper at ICLR 2026
Average Firing Rate by Layer
LIF FLIP
Average Firing Rate
84
posch enkeedd grog lif
patc, cmboell pooj lin
pacc umbedd proj lnP
paca, embeed lids
parc, umbhet pethollpppp...oo le
blodk (tamm lif
blodk: camu Ji
biook ant arte Jif
blodk unn grog Ji
block camil Ret Jin
blocks wes.ttmpppaa Roc Jif
blodd anna 3I
bled, altn Jrr
blod: amna JIr-
Mack alto anie DIr-
block Jir-
Mack mijuu jir-
Mackk mipin22 Jir-
Layer Name
Figure 12: Comparison average firing rates across layers for LIF and f-LIF models. The blue line corresponds to the LIF model, and the orange line corresponds to the f-LIF model.
where SOP denotes the number of spike-basse based accumulate operations.
For each layer the spike-based operation count is
SOPs(I) = p x T FLOOPs()-
(66)
Combining the above equations, the overall energy consumption can be expressed as
Exotal ESpiiFormer
Encuron
N Nacurons-
(67)
Here, N Nucurons denotes the total number of neurons in the network.
As shown Fig. 12, Table 16, and Table 17, our f-LIF neurons enable the network to achieve a lower average firing rate compared to the standard LIF node, although the fractional dynamics introduce additional computational overhead. This results in overall energy usage that remains at a comparable level, balancing the trade-off between energy efficiency and enhanced temporal modeling capabilities.
39

---

Published as a conference paper at ICLR 2026
Layer-wise energy accounting (SpikFormer). We report per-layer energy with sparsity- aware synaptic costs and neuron-intrinsie costs. "Energy Type" indicates whether the layer is counted with AC or Eac entries of the form "EAC r" denote AA cost scaled by the measured average firing rate r.
Layer Name
Type
TxFLOPs (M)
Average Firing Rate
Energy Type
Energy (mJ)
patch_ socch_emeed.ppe ptopddrrr_oooooooooooooo proj_coov conv
Conv2d
150.99
1.000
0.69
patch_embees emoj lif
LIFNode
0.00
0.378
Neuron
0.12
patch_emeee embed. proj conv1
Conv2d
1207.96
0.378
EAc 0.378
0,41
patch_embed.ppe projedif.oo_fl
LIFNode
0.00
0.318
Neuron
0.06
patch_embed.j Boceeeddpoo__oooooeooeeeoo proj_conv22 conv2
Conv2d
1207.96
0.318
Eac 0.318
0.35
patch_ biiheeebbdd.rgg proj_lif2 Jif2
LIFNode
0.00
0.396
Neuron
0.03
patch_ember embed. proj 3
Conv2d
1207.96
0.396
EΛc 0.396
0,43
patch_eeeees embed.proj lif3
LIFNode
0.00
0.413
Neuron
0.02
aach__ebbedrre_ccom
Conv2d
603.98
0,413
Eдc 0.413
0.22
patch_ patch_embed.rpe..
LIFNode
0.00
0,425
Neuron
0.00
block.0.attm..
Convld
67.11
0.425
EλG 0.425
0.03
block.0.attn.c lif
LIFNode
0.00
0.289
Neuron
0.00
block.0.attm. k_
ConvId
67.11
0.289
EAC 0.289
0.02
LIFNode
0.00
0.084
Neuron
0.00
block.0.atto.
Convld
67.11
0.084
EaG 0.084
0.01
block.0.atta.c
LIFNode
0.00
0.198
Neuron
0.00
block.0.attn
SSA
33.55
0.198
EΛC 0,198
0.01
block.0.attm.e lif
LIFNode
0.00
0.837
Neuron
0.00
block.0.att..pe
Convld
67.11
0.837
Eac 0.837
0.05
Block.0.attn.pret proj_lif
LIFNode
0.00
0.549
Neuron
0.00
ConvId
268.44
0.549
EAG 0.549
0.13
Block.0.mlp.fe11 lif
LIFNode
0.00
0.322
Neuron
0.00
block.0.mlp.fc2.
ConvId
268.44
0.322
EAC 0.322
0.08
block.0.mlp.fee2.
LIFNode
0.00
0.089
Neuron
0.00
block.1.attn.q.com
Convld
67.11
0.089
Eac 0.089
0.01
block.1.attn.q..
LIFNode
0.00
0.181
Neuron
0.00
block.1.att.. aat.attt....ovvv.
ConvId
67.11
0.181
EAC 0,181
0.01
block.1.att..k lif
LIFNode
0.00
0.062
Neuron
0.00
block.1.attn.y_com
Convid
67.11
0.062
Eac 0.062
0.00
block.1.attn...2
LIFNode
0.00
0.183
Neuron
0.00
block.1.attn
SSA
33.55
0.183
EAG 0,183
0.01
block.1.attn.tth lif
LIFNode
0.00
0.844
Neuron
0.00
block.1.attm.pre conv
Convld
67.11
0.844 EAC 0.844
0.05
block.1.attn.pre
LIFNode
0.00
0.485
Neuron
0.00
block.I.mlp.fel_com
Convld
268.44
0.485
EAG 0.485
0.12
block.1.mlp.fe11. lif
LIFNode
0.00
0.344
Neuron
0.00
1lock.1.mlp.fc2. block.1.mlp.ff22. head
ConvId
268.44
0.344
EAC 0.344
0.08
LIFNode
0.00
0.022
Neuron
0.00
Linear
0.03
0.022
EAG 0.022
0.00
Table 16: Layer-wise energy breakdown with sparsity. "Energy Type" uses FMAC for multi-
ply ply-accumulate energy and ÉAC scaled by the output average firing rate.
Esynaptic total 2.698,236.1 nJ,
(68)
Eneuron 235,486.99 nJ,
(69)
Eoverall 2,933.723.12 nJ 2.933723 mJ.
(70)
Energy composition:
Synaptic energy share: 92.0%.
Neuron energy share: 8.0%.
40

---

Published as a conference paper at ICLR 2026
Layer Name
Type
T'FLOPss (M)
Average Firing Rate
Energy Type
Energy (IJ)
patch_embed embed.proj
patch_
conv conv
Conv2d
150.99
1.000
EArAC
0.69
patch_embed embed.prof lif
f-LIFNecuro
0.00
0.362
Neuron
setth_embed.pree
Conv2d
1207.96
0.362
Eae 0.362
0.39
patch embed.proj lifl
J-LIFNcuron
0.00
0.315
Neuron
0.11
patch_
Btceeeddppoo__oooooeoeeeee proj_conv2
Conv2d
1207.96
0.315
Eac 0.315
0.34
patch_embed wtwpd.pooooooooo..o. lif2
f-LIINeurro
0.00
0.319
Neuron
0.05
seicc__eeed_jeee brobeeddrojjooooo proj_comv3
Conv2d
1207.96
0.319
Eac 0.319
0.35
patch_embed proj lif3
J-LIFNcuron
0.00
0.274
Neuron
0.03
conv
Conv2d
603.98
0,274
EAc x 0.274
0.15
_li
J-LIFNeuron
0.00
0.412 Neuron
0.01
block.0.attn.q
wak....ttt....oom conv
Convld
67.11
0.412
EAC 0.412
0.02
block.0.attn..
f-LIFNeuron
0.00
0,276
Neuron
block.0.artm.k conv
Conv Id
67.11
0.276
EAc 0.276
0.02
block.0.atte lif
F-LIFNeuron
0.00
0.091
Neuron
0.00
block.0.atto. W.t.O.att.yycoom
Convld ld
67.11
0.091
ΕaC 0.091
0.01
block.0.attn. v_lif
f-LIFNcuron
0.00
0.112
Neuron
0.00
block.0.attn
SSA
33,55
0.112 EAG 0.112
0.00
block...coo O.tttt.ttttnnn Jif
J-LIFNeuron
0.00
0.812
Neuron
0.00
block. block.0.attm.pro) p.t.ttttrooooocooo.. conv
Convld
67.11
0.812 EAC 0.812
0.05
wock.O.attn.pro) Jif
f-LIPNeuron
0.00
0,533
Neuron
0.00
block.0.mlp.ff _conv
Conv Id
268.44
0.533
FAC x 0.533
0.13
flock.0.mlp.fel1
f-LIFNcuron
0.00
0.315
Neuron
waokk..mll.fe22.coom
Conv
268.44
0.315
EaC 0.315
block.0.mlp..fe2_.t
J-LIFNeuron
0.00
0.034
Neuron
0.00
Llatt.L.attt.q...
Convld
67.11
0.034
EΛc 0.034
0.00
block.I.attn...1]
J-LIFNeuron
0.00
0.173
Neuron
0.00
block.1.att..4 c_nov
Convld
67.11
0.173
EAC 0.173
0.01
block.1.atto.k.
F-LIFNeuron
0.00
0.058
Neuron
0.00
I.ttk.I.attn.v_ee __covv
Cony ld
67.11
0.058
FAG 0.058
0.00
block.1.attn...ff
f /-LIFNeuron
0.00
0.101 Neuron
0.00
block.1.attn
SSA
33.55
0.101 EAC x 0.101
0.00
block.1.attn.att
f-LIFNecuro
0.00
0.826
Neuron
block.I.attt.prrj_ccoo croj
Conv
67.11
0.826
Eac 0.826
0.05
block.L.attm.rrg
F-LIFNcuron
0.00
0.471
Neuron
0.00
block.I.mlp.fel_.o
Convld
268.44
0.471
EAC 0.471
0.11
block.I.mlp.fel_Ii _if
J-LINNeuron
0.00
0.319
Neuron
0.00
block.I.mlp.fe22 conv block. bllpk.I.mlp.fe2_iii head
Convld
268.44
0.319
EAG 0.319
0.08
f- f-LIFNeuron
0.00
0.038
Neuron
0.00
Linear
0.03
0.038
EAC 0.038
0.00
Table 17: Layer-wise energy breakdown with sparsity. "Energy Type" uses ΣMAC for multi- ply- aly-accumulate energy and EAC scaled by the output firing rate for addition-only equivalents.
Esynaptic total 2,490,467.47 nJ,
(71)
F Fecuron total 421,324.24 nJ,
(72)
E Eoverall 2,9111,71.771 nJ = 2.911792 mJ.
(73)
Energy composition:
Synaptic energy share: 85.5%.
Neuron energy share: 14.5%.
41

---

Published as a conference paper at ICLR 2026
E f-SNN TOOLBOOX "SPIKEDE" PRESENTATION
We propose an open-source, out-of-the-box toolbox, named spikeDE, to support our f-SNN frame- work. built on the PyTorch platform. The toolbox enables the construction of SNNs through interfaces closely aligned with PyTorch. supports various neural network architectures like con- volutional neural networks (CNN). Transformer, ResNet, and multilayer perceptron (MLP) (Vaswani et al., 2017; LeCun et al., 1989; He et al., 2016; Zhou et al., 2022). We believe will serve the SNN community well, encouraging the advancement of broader class of SNNs that capture richer temporal patterns. The directory structure of the toolbox is as follows:
iiiit_
Package initialization
layer.py
# Base layer definitions
neuren.py
Base neuron definitions
odefunc.py
# E-ODE function definitions for SNNs
snn.py
# High-level SNN wrapper
solver.py
# E-ODE solvers
surrogate.py
# Surrogate gradient implementations
The neuron module provides variety of classic spiking neurons, including LIFNeuron (Leaky Integrate-and-Fing and IFNeuron Integrate-and-Fine along with unified interface for custom neuron dynamics. The snn module offers high-level SNN interface that supports either direct conversion artificial neural networks (ANNs) into SNNs or wrapping SNNs into trainable network objects.
SNN Neurons. spikeDE supports spiking neurons like LIFNeuron and IFNeuron. Besides, spikeDE supports custom spiking neurons. Users can define their own neuron types by inheriting from the provided BaseNeu ron class:
class LIFNeuron (BaseNeuron):
def forwarddse1f v_mem, current_innut-NNone) if perrent_input None: return m_m
tau = self.get_tau
dt 1.0
dv_nc_resect (-v_mem current_input) / tau y_post_charge v_mem dt dv_no_resct
spike www.roggte_ft___ssttte___e
- self. self.threshold,
self. suraoootrrggategradda.a dv_dt dv_no_reset (spike.detach sel tehr.thre)))ld)
return dv_dt, spike
class IENeuron
def (sel,, v_mem, current_innut-Noone) if current_input is None: return v_mem tau = self.get_tau( v_scaled v_mem self.threshold spike self. ((y_saledd self sturfssrrrgateeeg. sateee_rrr______ee_e áv_dt (-spike threshhre current_input) / tau return dv_dt, spike
Our LIFNeuron and IFNeuron produce two outputs: the first is the derivative of the membrane potential dv_dt, and the second is binary spike signal spike.
The surrogate_opt parameter specifies the surrogate gradient function used enable backpropagation through the non-differentiable spiking operation. Multiple options (e.g.. Carctan_suzrogat are provided, allowing users choose based on their needs.
How to Build and Train Custom SNN Network.
1. Building CNN-based Network The spikeDE framework is highly flexible and supports various
network backbones, including CNNs, ResNets, GNNs, and Transformers. By replacing the activation
42

---

Published as a conference paper at ICLR 2026
functions in traditional artificial neural networks (ANNs) with spiking neurons, spiking neural dynamics can be seamlessly incorporated into the model. Below is example using simple CNN backbone:
import torch.
class CustomCNN (nn.ncuull Modulc)
def _init_ (self, args):
super (CustomCNN, self). (aat)__inii init ()
Conv Blocks
self. Conv2d (2, 128, 3, 1, bias=Talse)
self.bnl nn. ww..Bhhhooodddd (128)
self.lifl LIFNeuron
args. tau, args. args.thresheld, args. srrasuuurroptte_gadscclllee
self.pooll (2.
self.conv2 nn. Cn.Con2d 128, 3, 1, bias=False) self.bn2 = nn. Nn.BBttchNorm2211255
self LIFNeuron
args. tav, args.threshcld, args. rrgs.surrogate__rrd_scss
self.poo12 nn.MaxPoc12d
self.conv3 Cn.Convd (128, 128, 3, bias=False) self.bn3 nn. Ba..aathoo22d self.lif3 LIFNeuron
args. tau, args.thresheld, args. surrrrogtttt_ggrasssllllle
self.pool3 (2.
self.conv4 nn.vond (128, 128, 3, 1, bias=False) self.bn4 = (aatchNorm2211255
self.1if4 LIFNeuron
args.tau, args.threshole args. serrssuuuoggaegeeeeeeeee scale
self.poo14 nn.MaxPoo12d
self.convs on.Conv2d 128, 3, bias=False)
self.bn5 mm.BBacchNor22d1125
self.lif5 LIFNeuron
args.tau, args. args.thresheld, args. sug.ssurrrooacccgra_ccolll
sclf.pool5 Mn.Maxool22d())
Fully Connected Layers self, flatten - nn. Flatten self dropoutt Dropuut (0.5)
self.ic1 In. (128 4, 512)
self.lif6 LIFNeuron args.tau, args.threshcld, args. suraogate_grra__scall
self.dropoute nn. an.Drrpuut (0.5)
self, fc2 = nn. Lineir (512, 110) gelf. LIFNeuron
tag, args. threshldh, args. args.suurroaate__raa_c.c..
Output Layer sel sel lay.pptttt__ votinglayerr(0) if args.vating else hear (110, 11)
43

---

Published as a conference paper at ICLR 2026
def (self, x)
# First convolutional block
self. (x)
self. (x)
x self.1ifl
x = self.pooll (x)
Second convolutional block
self. conv2 (x)
X se1f.bn2 (x)
X self.1if2
self.poo12 (x)
Third convolutional block
X sclf.conv3
sclf.bn3 (x)
sclf.lif3
self.pool3
# Fourth convolutional black
= selš.conv4
self.bn4 (x)
x selÉ.1if4
X self.poo14 (x)
Fi Fifth convolutional block
self.conv5 (x)
x self.bn5 (x)
x = self.1if5 (x)
X sclf.poo15 (x)
Fully connected layers
x self. flatten (x)
x self. f.dropoottt (x)
f (x)
felf.1if6(x)
x = self, delf.dropout2 (x)
self.fc2 (x)
sclf.1if7(x)
i Output layer
x = self, geelf.outut__layrre return
Users are free design other backbone architectures tailored their specific tasks.
Warpping Your Network. Once suitable network backbone is defined, can be wrapped into trainable SNN object using our SNNWrapper:
from spikeCE. import SNNWrapper
model ENNWrapper (CNNBackbone,
integratoo===deinn"" cattrppllation_methhoiee "lithodd=="iir"""" linear"
Set the initial input shape of the network (Needed!!!) secdel__sttnneura heeuoonsshapeeeeeeeeeeeeee kinput_shapees (1, 3, h, w)
h, w denctes the input image's height and width
44

---

Published as a conference paper at ICLR 2026
The resulting model is fully trainable SNN that be trained using standard PyTorch workflows, including automatic differentiation and backpropagation.
The imput to this object is tensor X RT>NN*, where T' denotes the number of time steps and N the batch size. The output O is a tuple containing the membrane potentials of each layer at every time step, as well as the accumulated spikes from the final layer.
The integrator argument supports two modes: "odeint" (for integer-order ODE integration) and " "ddiitt" (for fractional-order differential equation integration). Each integrator supports multiple numerical methods, allowing users to balance accuracy and computational efficiency.
3. Training your Network. Since our implementation is based on the Neural ODE framework, the training procedure differs slightly from standard PyTorch networks: time integration parameters can be specified or learnable. Below basic training loop.
def train (args, model, device, train_loadery optimizer, criterion):
"*"Train for one epoch. ***
model.train()
for batch_idx, (data, target) in enumerate (oader) cata, target data.to (device), target (device) data shape: IT, N, *1
Define input time points data_tine torch. (incpali
args.time_intest
(args. (args.time_steet stepsssssspssseeeee 1),
args.time_stepp
devico=device flcat()
method args. method Integration method, e.g., 'euler' atgs_step_siee
options - {
optimizent zero_grad
Define output time points (can be adjusted to save memory, especially with odeint odeint_adfoint) output_time = Norch.linspace! 0, args. sigsstimm_interval (args. 1),
args.T, device=device
fl
output model (data, data_time, output_ sime, methed, options) computes the mean of the output spikes # across the time steps in the output layer. loss criterion (outputtmman target) loss. backward))
optimizer.sten step
additional logging or evaluation logic
output... mean (0) corresponds to the network's 's final prediction across the time steps in the output layer, which can be used flexibly depending on the task (e.g., classification, regression).
F LIMITATIONS AND BROADER IMPACTS
F.1 LIMITATIONS
First, the hyperparameter tuning process for fractional dynamics, such as selecting the fractional order, can be non-trivial and requires domain expertise extensive experimentation. This may pose challenges for practitioners aiming to deploy the framework in new applications. Second, although our experiments show robustness noise and strong performance several datasets, the method has not been extensively compared with broader set recent graph learning methods in graph-structured settings. Further study is needed evaluate its practical constraints and comparative advantages
45

---

Published as a conference paper at ICLR 2026
against newer graph modeling approaches, including recent graph Transformer methods and graph optimization optimization-oricnted oriented frameworks (Tao et al., 2025; Zhao et al., 2025). The current fNNNent-SSNN toolbox also lacks mature distributed-trainnn modern toolboxes such as SpikingJelly implement CuPy-accelerated computation; integrating similar acceleration is promising direction for our framework.
support and remains under active optimization. In addition,
F.2 BROADER IMPACTS
The proposed f-SNN framework introduces biologically inspired approach to spiking neural networks, with the potential for significant positive impact on both research and application domains. By incorporating fractional-order dynamics, f-NN advances the modeling of complex temporal dependencies, offering new insights into brain-like computation and contributing to the understanding of non-Markovian behavior in biological neurons. This could inspire further interdisciplinary research in neuroscience and machine learning.
From application perspective, the ability of f-SNN to process temporal information with enhanced accuracy and energy efficiency makes well-suited for tasks such as edge computing, Internet of Things (IoT) devices, and neuromorphie hardware. The open-sourced toolbox provides prac- tical resource for researchers and practitioners, potentially accelerating innovation in fields like computational neuroscience, roboties, and bio-inspired artificial intelligence.
However, as with any machine learning framework, ethical considerations must be addressed. The deployment of FSNNN in sensitive applications, such as autonomous systems or decision-making tasks, should be carefully evaluated to avoid unintended consequences. Additionally, the increased computational requirements of fractional dynamics raise concerns about energy consumption during training, which may counteract the energy efficiency benefits of SNNs in some scenarios. Responsible use, combined with efforts improve computational efficiency, will be essential maximize the positive societal impact of this technology.
46


Reviews:
[{"review_id": "T7rhIOCx7V", "reviewer": "Reviewer_3jw6", "summary": "This paper proposes fractional-order spiking neural networks (s-SNN). The idea is to replace the linear ODE that describes the pre-spike potential dynamics of a (L)IF model with a fractional ODE. This allows for better capturing of long-term dependencies in the membrane potential evolution, enhancing the expressivity of the corresponding models. In particular, this paper shows how f-SNNs can be seen as a generalization of the (L)-IF SNNs and how to effectively discretize f-SNNs to get an implementable model using the Adams-Bashforth-Moulton discretization scheme. Finally, the author empirically demonstrates that f-SNNs consistently outperform traditional SNNs in both performance and robustness to input perturbations across multiple tasks and architectures. The authors further developed an open-source toolbox compatible with PyTorch, simplifying the design and implementation of custom SNN neurons.", "questions": "I ask the following questions to the authors:\n\n1) In the article \"Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Time\" the authors study the expressivity of the LIF SNN model both in terms of approximation properties and in terms of linear regions in the static case. Would you expect the f-SNNs neuron to produce more regions in the input space? Would you expect that f-SNNs neuron regions are in more general positions?\n\n2) Combining the fact that $\\alpha$ is selected through a hyperparameter tuning and that training SNNs in general is hard, what can you say about the overall efficiency of the training of an f-SNN model?", "limitations": "", "rating": 4, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well-written, precise, and enjoyable to read. The definition of the continuous-time f-SNN and the derivation of the discrete-time counterpart are well-explained. Also, the experimental setup is clear and well-documented, and the results are consistent with the claims.", "weaknesses": "The paper contains some minor imprecision that can be easily fixed. For instance, in equation (1) line 126, there is an a which I suppose is meant to be a 0. Moreover, in line 172, the authors mentioned \"learnable synaptic weights\" that are not introduced since in line 170, only the input current is mentioned $I_{in, k}$. The caption of Figure 4, page. 7, could be a little bit extended by explaining how to interpret Figure 4 (at least to me, it is not straightforward). \n\nI would appreciate seeing a related work section in the main part and a more detailed comparison with the existing models that exploit fractional derivatives. The authors mention them Appendix A2, but it is not clear how it compares with your model. This would be essential for the novelty assessment and the main reason for the overall score.\n\nThe theoretical result (Theorem 1 in Section 3.2) seems somewhat modest to be presented as a theorem, as it addresses only the constant input current case and relies on known solutions of the corresponding ODE and fractional differential equation. It might be more appropriate to present it as a remark.", "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": "ST0ZcmXObk", "reviewer": "Reviewer_xtnY", "summary": "This paper proposes a fractional-order spiking neural network (f-SNN) framework that replaces traditional integer-order ODEs with fractional-order ODEs to capture long-term dependencies in biological neurons, outperforming conventional SNNs in accuracy, energy efficiency, and noise robustness across neuromorphic vision and graph tasks while providing an open-source toolbox spikeDE for easy adoption.", "questions": "The energy consumption analysis of SNNs only includes synaptic analysis, while the energy consumption of the neurons themselves should also be compared.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The proposed fractional-order spiking neural network (f-SNN) framework effectively captures long-term dependencies in membrane potential and spike trains via fractional dynamics, addressing the limitation of traditional integer-order SNNs (based on IF/LIF models) that only rely on immediate past states.\n\n2. The f-SNN demonstrates superior performance in multiple aspects: it achieves higher accuracy than conventional SNNs on neuromorphic vision", "weaknesses": "1. The concept of fractional-order has already been proposed by others; the authors seem to have only put forward a framework and tested it on different datasets, which is more like an engineering problem.\n\n2. The datasets used are limited, and results on widely tested datasets such as CIFAR-10, CIFAR-100, and ImageNet are missing.", "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": "1Jrwk4hUM2", "reviewer": "Reviewer_XVNF", "summary": "This paper proposes a fractional-order spiking neural network (f-SNN) framework, which replaces traditional first-order membrane potential dynamics in SNNs with fractional-order ordinary differential equations (f-ODEs). The key idea is that fractional derivatives introduce non-Markovian memory effects, allowing neurons to integrate long-term dependencies following a power-law decay rather than exponential decay. The paper provides (1) theoretical analysis of fractional memory; (2) efficient discretization; and (3) extensive experiments on neuromorphic vision and graph datasets, showing consistent improvements in accuracy, energy efficiency, and robustness.", "questions": "1. How scalable is f-SNN to large datasets (e.g., DVS-Gesture full size) in wall-clock time and memory usage?\n\n2. Have the authors tried integrating f-LIF with surrogate gradient training, jointly optimizing α and synaptic weights?\n\n3. Can the same principle apply to recurrent  SNN architectures?\n\n4. Why do SpikingJelly and snnTorch produce such different results under the same dataset and network architecture?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The introduction of fractional dynamics into SNNs is novel and biologically inspired. It directly addresses the Markovian limitation of conventional LIF/IF neurons, linking to real neuronal behaviors and fractal dendritic structures.\n\n2. The derivation from Caputo derivatives to the discrete f-LIF formulation is rigorous. Theorem 1 and the comparison of exponential vs. power-law decay are clearly presented and intuitive.\n \n3. The f-SNN framework is a strict superset of conventional SNNs (α = 1), making it easily integrable into existing architectures and toolkits.\n \n4. Results across multiple domains (neuromorphic datasets and graph benchmarks) show consistent gains. The robustness tests (noise, occlusion, temporal jitter, etc.) are comprehensive and convincing.", "weaknesses": "1. The theory mainly discusses long-term dependence qualitatively via the Mittag–Leffler function; deeper analyses such as stability, gradient dynamics, or expressive power would strengthen the theoretical contribution.\n\n2. Although the paper mentions O(NlogN) and truncated-memory approximations, there is no clear empirical benchmark on training/inference speed or memory consumption compared to standard SNNs.\n\n3. Lack of ablation on fractional order $\\alpha$. While $\\alpha$ is a key hyperparameter, the paper does not analyze its sensitivity or interpretability across tasks.\n\n4. Some sections (e.g., Sec. 3.1 discretization) could benefit from clearer notation and step-by-step explanation. A schematic showing the memory kernel evolution over time would help readers intuitively grasp the effect of $\\alpha$.", "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)