Random Spiking Neural Networks are Stable and Spectrally Simple
Authors: Ernesto Araya
Massimiliano Datres
Gitta Kutyniok
Venue: ICLR 2026 Poster (2026)
Keywords: Spiking Neural Networks
Stability
Simplicity Bias
Random Networks
Primary area: learning theory

Abstract: Spiking neural networks (SNNs) are a promising paradigm for energy-efficient computation, yet their theoretical foundations—especially regarding stability and robustness—remain limited compared to artificial neural networks. In this work, we study discrete-time leaky integrate-and-fire (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classification tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum on low-frequency components. Motivated by this, we introduce the notion of *spectral simplicity*, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analysis to the *simplicity bias* observed in deep networks. Within this framework, we show that random LIF-SNNs are biased toward simple functions. Experiments on trained networks confirm that these stability properties persist in practice. Together, these results provide new insights into the stability and robustness properties of SNNs.

Published as a conference paper at ICLR 2026
RANDOM SPIKING NEURAL NETWORKS ARE STABLE AND SPECTRALLY SIMPLE
Ernesto Araya
Massimiliano Datres
Department of Mathematics, Maximillann--niveerre München Munich Center for Machine Learning (MCML) araya@math. 1mu. de
Department of Mathematics,
I
Universittt München Munich Center for Machine Learning (MCML) datres Gaath 1mu de
Gitta Kutyniok
Department of Mathematics.
Audwig-Maximilanm Universsitie München Munich Center for Machine Learning (MCML)
University of Tromsø
DLR-German Aerospace Center kutyniokémat h. 1mu de
ABSTRACT
Spiking neural networks (SNNs) are a promising paradigm for mergy-efficien computation, yet their theoretical foundations-esperte regarding stability and robustness-remain limited compared artificial neural networks. this work, we study discrete-time leaky integrate-and-Fn (LIF) SNNs through the lens of Boolean function analysis. We focus on noise sensitivity and stability in classifi- cation tasks, quantifying how input perturbations affect outputs. Our main result shows that wide LIF-SNN classifiers are stable on average, a property explained by the concentration of their Fourier spectrum low-frequency components. Mo- tivated by this, we introduce the notion of spectral simplicity, which formalizes simplicity in terms of Fourier spectrum concentration and connects our analy- sis to the simplicity bias observed in deep networks. Within this framework, we show that random LIF-SNNs trained networks confirm that these stability properties persist in practice. To- gether, these results provide new insights into the stability and robustness proper- ties of SNNs.
biased toward simple functions. Experiments
INTRODUCTION
Artificial Neural Networks (ANNs) have become central modern machine learning, but their rapid growth in scale demands increasingly unsustainable computational and energy resources (Thomp- son al. 2021). This challenge is especially acute for low-power devices, where efficiency is critical. Spiking Neural Networks (SNNs), inspired by biological neurons and operating through event-driven spikes, offer promising energy-efficient alternative (Roy al. 2019 Davies al.) 2018). Their sparse communication and compatibility with neuromorphic hardware position them
as strong candidates for sustainable machine learning (Mehonic al. 2024 Fono al. 2025).
Despite this potential, the theoretical understanding of SNNs remains limited compared to that of classical ANNss. While substantial progress has been made on designing training algorithms (Eshraghian et al. (202)) and hardware implementations (Davies et al. 20188 Indiveri & Liu 2015), core theoretical properties-su as stability, robustness, and generalization are still largely un- derexplored. This gap limits our ability to rigorously assess the strengths and limitations of SNNs in practice.
"Equal contribution.

---

Published as a conference paper at ICLR 2026
Among these properties, stability is crucial for designing networks resilient input perturbationn and adversarial attacks, yet no single definition exists. It can refer to algorithmic stability of learning algorithms (Elissceff et al.) 2005). dynamical systems stability (naturally aligned with the temporal dynamics of spiking neurons) (Ding et al. 2024), or-as we consider here-to sensitivity to input changes. Intuitively, a stable network should be resilient to small perturbations in inputs or parame- ters, which is critical for both reliable inference and efficient learning. Many SNN models, including the discrete-time Leaky Integrate-anc -ar-Fee (LIF) neuron, can be viewed iterative compositions of Boolean functions, since neurons emit spikes only when their membrane potential crosses a thresh- old (a binary event), motivating the study of stability via Boolean function analysis (O'Donnell, 2014). While prior work has examined SNN stability from dynamical systems (Ding et al., 2024) or neuroscience perspectives (Calaim et al. 2022), to our knowledge this is the first study to apply Boolean analysis to characterize and investigate stability in SNNs.
The Boolean perspective on SNNs suggests a natural link the notion of simplicity in neural net- works: if network is stable, small input perturbations rarely change its output, hinting bias toward "simple" input-output mappings. In the SNN setting, this can be formalized via the pre- dominance of low- cww-freuenccy components in the Fourier-Walk Walsh expansion of classifier. Motivated by this, we introduce spectral simplicity, which quantifies the concentration of the Fourier spec- trum on low frequencies. This notion connects naturally with other notions of simplicity studied in the context of simplicity bias, proposed as a lens to understand generalization in deep networks. For example, prior work has shown that random deep ANNs tend to implement "simple" functions (De Palma et al. 2019). formalized as a large average Hamming distance to the nearest input with different predicted class (Valle-Pérez et al. 2019). Spectral simplicity represents a weaker notion, but one that arises intrinsically in spiking networks.
To make these questions precise, we focus on the discrete-time LIF model. This model combines theoretical simplicity with widespread adoption in practice, serving as the foundation of software frameworks such as SNN-Torch and being implemented in digital neuromorphic platforms includ- ing Loihi 2 and SpiNNaker 2 (Orchard et al., 202T Gonzalez et al., 2023). In this work, we restrict attention to networks at initialization. This choice reflects the still- sill-developing theory of SNN train- ing and our goal of isolating stability properties intrinsic the model, without confounding effects from learning dynamics. Moreover, random networks have been shown to serve as useful priors in PAC- Bayes generalization bounds (Valle- Pérle- al.) 2019). Understanding the effect of training dynamics an exciting question; here, we explore it experimentally and leave a theoretical treatment for future work.
Contributions. We summarize our main contributions as follows:
We derive quantitative bounds on the stability of discrete-time LIF SNN classifiers, show- ing that they are stable on average with respect to random parameter initialization. In par- ticular, when input sequences lie in the binary cube {-1, the classifier output remains unchanged under perturbations of up to O ((v)) coordinates, with high probability for n large enough. The bounds depend explicitly on the model's hyperparameters and reveal that LIF SNNs exhibit stability properties comparable to other indepednde))))n)) Boolcan networks (Jonasson et al. 2023
We introduce the notion of spectral simplicity, defined via the Fourier decomposition of discrete-time LIF SNN classifiers for static data. We prove that random SNNs are biased toward spectrally simple functions, i.e., those whose Fourier spectrum is predominantly concentrated on fow-frequency components.
We complement these theoretical results with numerical experiments, investigating in par- ticular the effect of training stability. Our experiments reveal that both shallow and deep SNNs are noise stable, and that training tends to increase their stability on average.
1.1 RELATED WORKS
Stability of ANNs and SNNs. As previously noted, stability in SNNs can be approached from
multiple perspectives. Each neuron follows (typically non-linear) dynamical system, raising the natural question of how input perturbations affect its output. In the ANN and Neural ODE litera-
ture, Lyapunov-based analyses (Jidenez-Rodriguez et al. 2022 Kang et al. 2021 Rahnama et al.)
2

---

Published as a conference paper at ICLR 2026
establish robustness via bounds on how perturbations propagate through the network. The discrete-ting titrete-timeeeee LIF model's 's robustness to input perturbations has been analyzed from a dynamical systems perspective by Ding al. (2024), who derived bounds on output spike sequence variations under perturbed inputs. Our work differs in three key ways: (i) we study stability at the classifier level, where predictions may remain unchanged even if spike trains differ; (ii) consider the reset- by-subtraction mechanism, which introduces additional complexity compared to the reset-to-zero simplification in (Ding et al., 2024); and (iii) our analysis extends beyond single neurons multi- neuron networks. Methodologically, our approach is different building on Boolean function analysis [O'Donnell] 2014). aligning more closely withsJonasso et al. (2023). but introduces new challenges stemming from reset dynamics and temporal evolution, which create nontrivial probabilistic depen- dencies.
2019)
Simplicity bias. Simplicity bias has been proposed as mechanism underlying the generalization of deep networks, both in trained models with SGD (Arpit et al., 2017: Nakkiran et al. 2019] Valle- Pérez et al. 2019) and in random ANNs (De Palma et al. (201)). The central idea is that learning favors simple functions, though the precise definition of simplicity varies, and pitfalls have been noted in (Shah al. 2020). We introduce spectral simplicity, based the
Walsh decom- position of Boolean functions, as a complementary notion. While our focus on random networks parallels (De Palma at. 2019), the techniques differ substantially: their analysis relies Gaussian process arguments, whose extension to SNNs is unclear, whereas we draw on Boolean function anal- ysis, which adapts naturally to our setting. While weaker measure of simplicity, spectral simplicity arises organically in spiking networks.
1.2 NOTATION
Give a positive integer a, we denote with [a] the sets {1.....a} Given x, y E (-1,1}% we define the Hamming distance between x and y as des (It, y) = [ [m] oa # )1 1 Σ Jay vsl- We use sign(x) E {} for the sign function, i.c. sign(x) = 1 if 2 0 and -1 otherwise. We write N N(u, Σ) for the Gaussian distribution with mean and covariance E, (1i,(((1,1)") for the uniform distribution
the hypercube, and Bin(n, p) for the Binomial distribution with parame- ters (n,p) Similarly, Rad(x) denotes a vector with i.i.d. Rademacher coordinates with parameter x(dimension clear from context). For asymptoties, we use the standard asymptotic notation: O(-) o()) and w()).
2 THE DISCRETE-TIME LIF MODEL.
We aim to study the stability of SNNs in classification tasks. Specifically, we consider SNN models constructed compositions of sign leaky integrate-and-fiee (sLIF) neurons. Each neuron acts as an information information-processinng unit that maps time series inputs (x+ (#I))E(T] E (f (f"1, to binary spike sequences (St)err E 13,, based on a neuronal dynamic. The dynamics of a single neuron are defined as follows.
Definition 1 (sLIF neuron). Let T 2 1 be an integer, 3 E [0,1]. 0 E (0). We define the sign leaky integrate-and-fii (sLIF) neuron, with input (Zt)tE|T)
(f-1,1)myy and output (st)ett)) E (1-1,13)* as a parametric computational unit that evolves over discrete time steps t [T) accordingly to the following recursive dynamic
Ut = Buc-1 + ww Øe ((--1+1)
8, sign
U1 -0)
ug
where (u.)em E (10, T weights. Equivalently, this can be regarded as function mapping (It)tETT) + (St) To make the dependence on inputs and parameters explicit, we occasionally use the notation 8t ((zx) w..
is the sequence of membrane potentials and w E R² are the model's
While theory focuses on hinary data, the definition extends real-valued inputs.
3

---

Published as a conference paper at ICLR 2026
Each neuron processes input sequence (21) through its membrane potentials (u4 te(j, which evolve according to an autoregressive decay dynamics over the time horizon T (referred as the latency of the model). The leak parameter [0. 1] controls this decay, specifying the fraction of potential retained per time step. Whenever the membrane potential exceeds the activation threshold time f', the neuron emits spike, recorded 8t 1 in the spike train (8t)aE(T)
0 0 at and the potential is reduced by (reset by subtraction). Weights initialized ensuring w x = O(1) with high probability for x E 1,", thereby avoiding degenerate regimes of vanishing or overly frequent firing. For further background the LIF model, see (Gerstner & Kistler 2002). In short, we use the term leaky mntegrate-and-Poe (LJF) neuron for the variant with a Heaviside step function and reset rule U, + V2- 0 1: and integrate-and-Fi (IF) when the leakage is absent (3 1); networks composed of such units are referred to as LIF and IF SNNs, respectively.
~ (O I, /n),
Networks of spiking neurons. the sLIF SNNs considered here, multiple sLIF neurons are interconnected through weighted synapses. The spike train generated by each neuron can serve as input to other neurons, and the overall network dynamics evolve over time, akin to a recurrent network. In what follows, we focus on SNNs composed of fully connected sLIF neurons, arranged in layers, as formalized the next definition.
1, E 0,, 0 E [0, 20), and let the imput sequence be (xT))e(T) E ([1,11)m)) An I.-layer sL.F neural network with latency T and layer widths nL is defined the Boolean dynamical system
Definition 2 (sLIF SNN). Fix positive integers L,T, T.T,no
(1) Bu + (1l) ( +1),
( sign (ut 0),
81
(2)
(1) O, u0
8; Zt.
where, for each layerl E [L]. W E R denotes denotes the weight matrix, (u, )ee the membrane potential sequence, and ($) (1)
Deeryy is the (output) the spike sequence. We collect all parameters into a single vector W = v (W(1).. (....,(()) Rdd, where
Given I--layers sLIF SNN as in Definition a widely used choice of classifier in SNNs (see e.g., (Dichl & Cook 2015)) is based on spike counts at the output layer, namely
(1:)(( W) arg max AE (FZ)]
(L) 81,
((Ie))E))) W),
where the predicted class corresponds to the neuron in the final layer with the largest total spike (l. count. Here, 8/2 is the i-th coordinate of the spiking sequence s(4).
Assumptions. Throughout this work, we assume no the number of classes. The extension of the subsequent results the case * discussed in Appendix D For simplicity, we focus analysis allows dynamic input sequences (Zt)ee(r) though some results only apply to static inputs, interpreted repeated presentations of the same sample over time. Such constant input encoding commonly used practice for time-static datasets such as MNIST or CIFAR-10 (Rathi & Roy
7T--1
with n,. equal to
variant of the sLIF model equation 2 with 8 1. Our
2023 Rueckauer al. 2017). Finally, the model parameters initialized as
W, i.idd ((0,1/n), i E [d].
3
NOISE SENSITIVITY OF BOOLEAN FUNCTIONS
Within this framework, sLIF neurons can be viewed as compositions of Boolean functions, allowing stability analysis via Boolean function theory. We recall the basic definitions here and refer the reader to Donnelll 2014) for a comprehensive treatment. Additionally, we introduce spectral simplicity, which is central to the statement of main result.
4

---

Published as a conference paper at ICLR 2026
Noise sensitivity and stability. The stability of Boolean function is classically quantified by its noise sensitiviry. For f [-1,1]" + {-1,1} and noise rate E [0, 1], define
Nf) (x.;(f()) + f(z G E)). where 1n,,((-111)") and (f1,...,,.)) has i.dd entries &, Rad(1 - v). Equivalently, the noise stability is
Stab)-2 E.,e ( o 6)] 1 -1-2NS,
capturing the probability that f preserves its value under input perturbations. Given this equivalence, we will use noise sensitivity in the sequel.
Definition (Expected noise sensitivity). For parametric famity hew. a probability measure
on W, and x, & distributed as above, define
ENS. (1S bw-p)) == IP a <l # (x ④ E)]-
Fourier analysis and spectral concentration. Every f {-1,1}" + R admits unique Fourier-Walsh expansion (O O'DDnnell 2014 Thm. 1.1):
f(x) = Σ x(()), Xs(x) = II 24. SCIm
iE
Low-degree terms ([S) small) correspond low frequencies, and high-degree terms high fre- quencies. A constant function, for instance, has spectrum supported on 0. The notion of spectrum concentration formalizes when a function is "simple": most of its Fourier weight lies on low-degree terms, i.e., subsets of size o(n). Formally (O'Donnell 2014 Def. 3.1), is e-concentrated up to degree
Σ A(S)² S
S/S
Definition 4 (Expected spectrum concentration). Given a parametric family of functions (Jo {-1,1]" - R wew. and a probability measure μ in W, we say that (Iw hew has, in expectation under μh spectrum e-concentrated up-to degree if
E
Ewery
Σ P(S() SC SS
S.
A key connection between noise stability and spectral concentration is given by (O'Donnell 2014 Prop. 3.3), which states that for a Boolean function f. if we set NS. then the spectrum of f is e-concentrated up degree 1/v. This result extends naturally parametric families of func- tions by linearity, replacing NS.(J) with ENS,()) and spectral concentration with its expected counterpart (see Lemma
Other notions of simplicity. Alternative notions of simplicity have been proposed in the sim- plicity bias literature (Valle-Péres al.) 2019 De Palma (019). Closest our setting is the definition of De Palma et al. (2019), who study, for a classifier f [-1,1]" + {-1,1} and E 11. the quantity
(x; = {y (r, u) = h, f(x) + f(y)
(4)
which counts h-bit perturbations that flip the label. They show that its expected asymptotic behavior of depends on the activation function; for ReLU networks (it--uni(((1,1)))))b
f)] + 0, imply-
ing that the average Hamming distance to the nearest input with a different class is O log connect this notion with noise sensitivity, note that for any parametric family of Boolean functions {f new.
v ). To
ENS, (1Sw)) m-p)) )
Σ Eurppa- (N (w)) (0 -0) v)" n-h
h=1
In principle, could recover E[N, (2; fww] by inverting the previous relation, but this would quire a precise characterization of ENS,. In practice, only bounds on this quantity are typically attainable, show next.
5

---

Published as a conference paper at ICLR 2026
4 NOISE STABILITY OF SNN CLASSIFIERS
In this section, we apply the Boolean function analysis framework from Section 3 to quantify the stability of discrete-time LIF-SNN classifiers, Recalling the SNN classifier definition in equation and our main assumptions from Section 2
we begin with the single neuron case.
Single neuron stability. Under Definition | each output 8, in the sequence (st) defines Boolean function 5t {-1, + (ge) (e()) with fixed Hamming distance, we bound the probability (over random initialization) that si (Z()))))))) w) # Su(()))E)() w.)
[-1,1]. For any & E [T]] and input sequences (Zr))e(r)) and
Theorem 1. Consider sL.F neuron, with latency T E N,, threshold O E (0, 00), with random parameter vector w0,N00 In/n)- We consider two imput sequences T1,-..,, IT E 1- and and V. Ex= Vs. mf De = 0
, E {-1,1}". Denote V = ( (x),)))) Vo)/n
then for all E [T), we have
IPoo ($1(x11..... s1) + selyi........ 3.)1 S C(1 + 032 Vor
where C O is absolute constant independent of the 0, T, t, 21:-.., PT and yi..... yT- Moreover, for static inputs the same bound applies without the logarithmic factor.
Below we present proof sketch of Theorem crreorrmm[occutiiie focusing on the primary technical challenges; the full proof is deferred to Appendix C.I For the case with 3 * 1, and with Heaviside activation instead of sign, we refer the reader to Appendix D
Proof sketch of Theorem We argue by induction. Here, illustrate the argument for 1. In this case, the problem reduces to bounding the probability that a random linear threshold function produces different outputs for two inputs 21, 8/ Hamming distance |//n|:
sign(w x1 - 0) * sign(w" y1 0)
Step 1: Decomposition. Define X = w æ1 and Y = w y1. which are standard Gaussians. A classic Gaussian decomposition property allow us to express Y pX + /1 - Z² where - 1/1
Step 2: Event characterization. The disagreement event is equivalent to
(X> 0, Y < 0) U (X <
0}.
Step 3: Probability estimate. The first event has probability
P 0, S = 226 (020-0,0 0; 2n1 - 1))
where Φ denotes the bivariate Gaussian CDF with unit variances and correlation 2 1. The second event yields a symmetric expression 20 0; 2v1 1). Combining both, and using tail bounds and Lemma one obtains
PP[sign(w" Tx 0) + sign(w (w 0)} < Ca VPT,
for a constant Ce depending only on 0, see equation 10 for details.
For 2 2, the argument extends but with added complexity: temporal dependencies require union bounds, introducing the 7' factor in the constant. In the dynamic case, the sLIF neuron processes input sums that lie outside the hypercube, creating technical challenges. In contrast, the static case avoids these issues and the proof is simpler, with better rates.
Remarks. For T 1, our result recovers (up to logarithmic factors) the known bounds on the noise sensitivity fixed linear threshold functions, i.e., classifiers of the form sign(w' w x 8). For larger T, our bounds deteriorate. This may partly reflect proof coiffactssince handling dependen- cies introduced by shared weights across time can loosen bounds- but is also consistent with the general behavior of Boolean function compositions, where sensitivity typically increases with depth.

---

Published as a conference paper at ICLR 2026
A key obstacle to sharper time dependencies is the reset mechanism: thresholds adapt dynamically as the process evolves, complicating tighter analysis.
By inspecting the proof, we note that the assumption & 1 can be relaxed without significant changes to the analysis, but we maintain it here for simplicity. The proof also applies, with minor modifications, to the classical LIF model with Heaviside activations, but adopt the signed variant to enable a cleaner Fourier analysis. contrast, the requirement of large network width is essen- tial, as our concentration-bant stability is explored empirically in Section S
arguments rely on it. The influence of architectural parameters on
Multiple neurons. We now analyze the noise sensitivity of classifier outputs from I-layer sLIF neural networks. Specifically, we study classifiers defined as in equation 3 and built from L-layer sLIF networks (Definition 2)
Theorem 2. Let fL.T(. W) be a nla-ssssses classifier, defined by a L-layer SLIF, according to equation 3 with latency T E N4, 0 (0),)), widths nı Y22 = AL-1 nu and weights W N (0, Ia/n). Let ;1,..,, E [-1,1]" and "1:...., yr [-1,1}" such that ( 9t) [dnn] with E [0, 11. Let us define v == m V4 and assume V = O va Then, for n large
enough, holds that
IPw ((L)(())) W) + p4. (W))))))))
(001+++)00 S InrT + O)2 pلT log 3/2 n + (L - 1)e -w 2ad--11
for some absolute constants c, C independent of 0, T, n, t, $1....y , or and y...... yr.
We proceed by induction on 1 [L]. To illustrate, consider 1. Following (Jonasson et al. 2023 we analyze the Markov chain
Proof sketch of Theorem
D (xn, vn) 11
(m)) (m)) 1" (w))
E [T,]
which has n + states and absorbing state O. Conditioned on ((--1)= [unn]. we have
(x. ~ Bin(n, P,,):
S Covvn
where the bound on Pv, follows from Theorem Hence D (z. y) is stochastically dominated by Bin(n, Co (P)) which leads to the desired bound via Chernoff bounds (see Theorem 3 and standard manipulations. For 2 2, we repeat the previous argument which involves repeteadly applying Chernoff bounds see Appendix D.3
The following corollary links the probability that input perturbations change the output to a bound on the expected noise sensitivity of binary SLIFF-NN classifiers. This, in turn, characterizes their spectrum and quantifies the degree spectral simplicity introduced in Section 3 The proof is deferred Appendix D.4
Corollary 1. Let (', W) be binary classifier (nz 2) in Theorem and assume static inputs. Then, for any v' S Vologn we have, n large enough,
ENS, (LLL. W) W)) S CT ov golet 3/2
(L 1)e-or te-
where CT > 0 is constant, as in Theorem 2 Moreover, for sufficiently large n, the family of L-layer sIIF SNN binary classifiers has, expectation under (0, Ta/n), spectrum e-concentrated (Definition 4p up to degree 1 1/v', with
= Or 20 1og
3/2 n.
Vnlogn In this case, an L-layer binary SNN classifier is (211/2(01)) concentrated up degree ( (V log Thus, only vanishing fraction of degrees contribute meaningfully to the spectrum, making these classifiers spectrally simple. Interestingly, the bound the maximal degree of concentration is independent of architectural parameters, while
Remarks. To illustrate, take v/

---

Published as a conference paper at ICLR 2026
the concentration level deteriorates with larger L, T. and 0 (the log"/2 seems artifact of the proof). The increase with L is not surprising, since compositions of Boolean networks typically behave similarly, and analogous results are known for threshold ANNs (Jonasson et al., 2023). The O-dependence appears to be proof artifact. Whether the T- and I--depenenncies are intrinsic re- mains open question, which we investigate experimentally in the next section.
5 NUMERICAL EXPERIMENTS
We empirically evaluate the noise sensitivity, ENS,, of various spiking neural networks to inves- tigate how our proposed simplicity measure relates to the model dimension n and the tightness of the bounds established in [ and in the case of static inputs. In addition, we study how training (signed) SNNs influences their sensitivity to random perturbations of the input.
Single LIF experiments. We approximate the noise sensitivity ENS for different IF and sIF spiking neurons with imput dimensions n = 100, 1000, 10000, threshold 0 0.5 and Specifically, we compute a Monte Carlo approximation by uniformly sampling 10 model's 's weights, 100 data points, and 100 random perturbations. The results are shown Figure la and 16 We observe that, for all considered the neurons exhibit low sensitivity for both sIF and IF models. In conclusion, we stress that even though for our result we need the signed neuron versions, the sensitivity seems to be similar for the IF neuron. Moreover, the theoretical bound proved in Theorem iss satisfied both for the sIF and the IF neuron.
10.
Additional experiments for different values of v, 0.5, and 0 0 can be found in Appendix E see Figure Figure 6 and Figure respectively.
Expected sensitivity
Expected sensitivity
ENSu=
ENS
04-
Input Dimension
Input Dimension (n)
(a)
(b)
Figure 1: Noise sensitivity ENS1/V for different input dimensions for sIF and IF neurons with 0 0.5 and T = 10. (a) sIF neuron (log-scale x-axis); dashed line: scaled bound from Theorem
(b) IF neuron (log-scale x-axis); dashed line: scaled bound from Theorem
IF SNN with 5 layers. We extend the experiments to the deep setting by considering and sIR spiking neural networks with five layers, using the same Monte Carlo approximation procedure as in the shallow case. We evaluate ENS, for input dimensions n = 100, 1000, 10000, with each layer having width equal to the input dimension, 0 0.5 and T 10. The results shown in Figure [a and 2b Although depth appears to have a stronger impact on sensitivity than latency, the bound presented Theorem 2 tends to overestimate the effect. Additional experiments for different values of y and 3 = 0.5 can be found in Appendix
Noise sensitivity after training. We evaluate the noise sensitivity of trained sIF and IF SNNs in static data. In particular, we train three-layer sLIF and IF SNN on MNIST (i.c., n = 784) using the ADAM optimizer with surrogate gradients (Eshraghian accuracy. After training, we estimate the output sensitivity by perturbing each test sample 100
2023) until reaching 98% training
times, flipping each component with probability v E [ 2 .... -े and measuring the fraction of perturbations that change the network's output. For comparison with random network, the noise
8

---

Published as a conference paper at ICLR 2026
Expected noise sersiiy of S-layers sIF SNNS at v
Expected noise sensity of 5-layers SNNs at
0.30
0.22
0.36
ENS..>
0.34
+5
0.15
0.32
ENS
0.10
0.50
0.12
0.18
0.0
3 Input dimension (n)
Input dimension (n)
(a)
(b)
Figure 2: Noise sensitivity ENS 1/ 1/ for different input dimensions for 5-layers sIF and IF neural networks with = 0.5 and T = 10. (a) sIF neuron (log-scale x-axis); (b) IF neuron (log-scale x- axis).
sensitivity is estimated using the same procedure, averaged over 10 independent random weight initializations. As expected, training significantly reduces the sensitivity of the model whenever the final test accuracy is sufficiently high. Figure [a shows show the results for MNIST. Figure shows the same experiment on CIFAR-10 (n 3072). Notice that training reduces the model's sensitivity, but less strongly compared to MNIST. This aligns with the fact that both the training and test accuracies are larger for CIFAR-10 (which achieves 84.38% training accuracy).
SNNs MNIST
v CIFAR10
0.6
0.45
Tained CIRRR1
Random FAR1)
0.4
0.50
EN5
Randam Mk.5T
Tranned 19A5T
0.25
0.15
0.1
0.10
0.000
0.005
0.010
0.055
0.000
0.075
0.003
0.055
0.000
c.005
0.00
0:015
0,070
3.005
n.bib
0.035
(a)
(b)
Figure 3: Sensitivity input perturbations in SLIF-SNNs (T' = 100, 0= 0.5, = 1, 3), shown at initialization and after training on (a) MNIST and (b) CIFAR-10.
Noise sensitivity on neuromorphic dataset. We evaluate the noise sensitivity of spiking con- volutional network on NMNIST Orchard et al.] (2015). NMNIST an event-based version of MNIST recorded with dynamic vision sensor. Each data point is tensor in {0, 1] xxxxxx34X34 where T ~ is the time, 2 is the polarity and 34 are the width and the height. We perturb the input sequence by replacing each time frame by flipping each component with probability
1 nin ...., vn where n = 2312(= 34 x 34 x 2). The results are displayed in Figure Notice that the model has small ENS both before and after training. Moreover, the training does affect the ENS as much as for static data.
E
Dropping 5% of the input. We evaluate the noise sensitivity when the 5% of the input component are randomly dropped. In detail, we evaluate the noise sensitivity for 5-layers (s)SNN with input dimension n = 1000, and we compare the probability of observing different output with respect to this input perturbation. The results are reported in Table We notice that, both for the LIF and the sLIF model, dropout leads to smaller output sensitivity than random flipping. This is indeed expected since dropout does affect zero components in the case of inputs in the (0, 1)" hypercube.
9

---

Published as a conference paper at ICLR 2026
ENS of LIF SNNs on NMNIST
0.175
Trained NMNIST
ENS
0.100
0.075
0.350
0.0000 0.0025 0.0050
0.0075
0.0100
0.0125
0.0150
0.0175 0.0200
Figure 4:
Sensitivity to input perturbations of convolutional SNN with T 100, 0.55 and 0.5, shown at initialization and after training on NMNIST
Sensitivity of 5-layer LI) neural networks with = 0.5 and under two types of input perturbations: random flipping and dropout. For each model, the lowest (i.e., most robust) sensitivity value is highlighted in bold.
Table I:
Model / Perturbation
Random Flipping
Dropout
sLIF
1.19
0.16
LIF
0.28
0.16
CONCLUSION
In this paper, study the stability of wide SNN classifiers through the lens of Boolcan function analysis. We provide quantitative bounds on their expected noise sensitivity and show how these stability guarantees connect to simplicity bias, motivating new notion of simplicity. We empirically validate Theorem [ in the case of single sIF and neuron. We show that both shallow and deep (s)IF neural networks exhibit small noise sensitivity in practice and this property extends also random dropout perturbation of the input signal. Furthermore, we empirically suggest that training tends preserve even improve the ESN of SNNs.
The classifiers we analyze are widely used in practice, and most of our assumptions can be relaxed. The main restriction is the requirement of sufficiently large widths, needed apply concentration of measure; whether this condition can be weakened remains an open question. On the other hand, the uniform input distribution assumed in Definition does not affect Corollary making extensions to other input distributions straightforward.
Future directions.
Several avenues merit further investigation:
Extending our results feedforward RNNs, where the dynamics are simpler, as well to more general non-feedforward SNNs.
Studying stability under alternative perturbation distributions, a step toward understanding adversarial robustness.
Investigating the average distance to the nearest input with difffrent label (see equation 4 as (De Palma et al. (2019). Unlike their setting, a simple union bound fails here, since our ((1/ bound is insufficient given the exponentially many inputs distance O(yn).
Understanding how initialization impacts stability in SNNs, and whether classical ANN initialization schemes are optimal this context.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENTS
Ernesto Araya, Massimiliano Datres and Gitta Kutyniok acknowledge support by the Munich Center for Machine Learning (MCML).
Ernesto Araya and Gitta Kutyniok acknowledge support by the German Research Foundation under Grant DFG-SPP-2298.
Massimiliano Datres and Gitta Kutyniok acknowledge support by the project "Next Generation AI Computing (gAIn)", funded by the Bavarian Ministry of Science and the Arts and the Saxon Ministry for Science, Culture, and Tourism as well as by the Hightech Agenda Bavaria.
Gitta Kutyniok is also grateful for partial support from the Konrad Zuse School of Excellence in Reliable AI (DAAD). Furthermore, she acknowledges support by the German Research Foundation under Grants KU 1446/31-1 1-1 and KU 1446/32-1.
ETHICS STATEMENT
This work focuses on the theoretical analysis of robustness in machine learning and does involve experiments on human subjects, sensitive personal data, or applications with direct societal risks. The datasets referenced are publicly available, and private or restricted data used. Potential ethical concerns related to misuse are minimal, as the contributions are mainly theoretical.
Acknowledgment of LLM Use. We explicitly acknowledge that large language models (LLMs) were used solely for polishing code, improving sentence clarity, and refining grammar. They were not used for generating research ideas, proofs, or results.
REPRODUCIBILITY STATEMENT
We have taken multiple steps to ensure reproducibility of our results. All theoretical claims are accompanied by rigorous proofs, presented in detail in the appendix. Assumptions underlying the theorems are explicitly stated, and definitions are given in full to allow independent verification.
REFERENCES
Devansh Arpit, Stanislaw Jastrzebski, Nicolas Ballas, David Krueger, Emmanuel Bengio, Maxin- der S. Kanwal, Tegan Maharaj, Asja Fischer, Aaron C. Courville, Yoshua Bengio, and Si- mon Lacoste-Julien. closer look at memorization in deep networks. In Proceedings of the 34th International Conference on Machine Learning (ICML), pp. 232-242. PMLR, 2017. URL
Nuno Calaim, Florian Dehmelt, Pedro J Gonçalves, and Christian Machens. The geometry of robustness in spiking neural networks. eLife, 1:e77776, may 2022. ISSN 2050-084X. doi: 10.7554/eLife.73276 URL httphtht//o....... org/ (1555////ff// 73276
Mike Davies, Narayan Srinivasa, Tsung-Han Lin, Gautham Chinya, Yonggiang Cao, Sankar Cho- day, George Dimou, Prasad Joshi, Nabil Imam, Shweta Jain, Yoonseok Liao, Chit-Kwan Lin, Andrew Liu, Deepak Mathaikutty, Stephen McCoy, Arnon Paul, John Tse, Rajeev Venkatara- manan, Andreas Wild, Yangi Yang, and Hong Wang. Loihi: A neuromorphic manycore processor with on-chip learning. IEEE Micro, :8(1))82-44 99, 2018.
Giacomo De Palma, Bobak Toussi Kiani, and Seth Lloyd. Random deep neural networks biased towards simple functions. In Advances in Neural Information Processing Systems, volume 32, pp. 13583-13594, Curran Associates, Inc., 2019.
Peter U. Diehl and Matthew Cook. Unsupervised learning of digit recognition using spike-timing timing- dependent plasticity. Frontiers in Computational Neuroscience, Volume 9 2015, 2015. doi: 10.3389/fncom.22 201.00000....
11

---

Published as a conference paper at ICLR 2026
Jianhao Ding, Zhiyu Pan, Yujia Liu, Zhaofei Yu, and Tiejun Huang. Robust stable spiking neural networks. In Proceedings of the s1 Interaational Conference on Machine Learning, ICML'24. JML,.org, 2024.
Andre Elisseeff, Theodoros Evgeniou, and Massimilianoo Pontil. Stability of randomized learning algorithms. J. Mach. Learn. Res., 6:55-79, December 2005. ISSN 1532-4435.
Jason K. Eshraghian, Max Ward, Emre O. Neftei, 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, 1110)):016-1154, 2023.
Adalbert Fono, Manjot Singh, Ernesto Araya, Philipp C. Petersen, Holger Boche, and Gitta Ku- tyniok. Sustainable ai: Mathematical foundations of spiking neural networks, 2025. URL https://arxiv.ort
Wulfram Gerstner and Werner M. Kistler. Spiking Neuron Models: Single Neurons, Populations, Plasticity. Cambridge University Press, 2002.
Hector Andres Gonzalez, Jiaxin Huang, Florian Kelber, Khaleelulla Khan Nazeer, Tim Hauke Langer, Chen Liu, Matthias Aleander Lohrmann, Amirhossein Rostami, Mark Schöne, Bern- hard Vogginger, Timo Wunderlich, Yexin Yan, Mahmoud Akl, and Christian Mayr. SpiNNaker2: A large-scale neuromorphic system for event-based and asynchronous machine learning. In Ma- chine Learning with New Compute Paradigms, 2023.
Giacomo Indiveri and Shih-Chii Liu. Memory and information processing in neuromorphic systems. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, 103(8):1379-1397 2015.
Ivan Jimenez-Rodriguez, Aaron D. Ames, and Yisong Yue. Lyanet: A Iyapunov framework for training neural odes. In Proceedings of the 39th International Conference on Machine Learning (ICML), volume 162 of Proceedings of Machine Learning Research, pp. 10234-10254. PMLR, 2022.
Johan Jonasson, Jeffrey E. Steif, and Olof Zetterqvist. Noise sensitivity and stability of deep neural networks for binary classification. Stochastic Processes and their Applications, 165:130-167, 2023. doi: 10. 20116f.spa.2023.08.0.m
Qiyu Kang, Yang Song. Qinxu Ding, and Wee Peng Tay. Stable neural ODE with lyapunov-stable cquilibrium points for defending against adversarial attacks. In A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan (eds.), Advances in Neural Information Processing Systems, 2021. URL
Adnan Mehonic, Danicle Ielmini, Kaushik Roy, Onur Mutlu, Shahar Kvatinsky, Teresa Serrano- Gotarredona, Bernabe Linares-Barranco, Sabina Spiga, Sergey Savel'ev, Alexander G. Balanov, Nitin Chawla, Giuseppe Desoli, Gerardo Malavena, Christian Monzio Compagnoni, Zhongrui Wang, Joshua Yang, Syed Ghazi Sarwat, Abu Sebastian, Thomas Mikolajick, Stefan Sle- sazeck, Beatriz Noheda, Bernard Dieny, Tuo-Hung (Alex) Hou, Akhil Varri, Frank Brüicerhooff- Plückelmann, Wolfram Pernice, Xixiang Zhang, Sebastian Pazos, Mario Lanza, Stefan Wiefels, Regina Dittmann, Wing H. Ng, Mark Buckwell, Horatio R. J. Cox, Daniel J. Mannion, An- thony J. Kenyon, Yingming Lu, Yuchao Yang, Damien Querlioz, Louis Hutin, Elisa Vianello, Sayeed Shafayet Chowdhury, Piergiulio Mannocci, Yimao Cai, Zhong Sun, Giacomo Pedretti, John Paul Strachan, Dmitri Strukov, Manuel Le Gallo, Stefano Ambrogio, Ilia Valov, and Rainer Waser. Roadmap to neuromorphic computing with emerging technologies. APL Materials, 12 110):109201, 2024.
Preetum Nakkiran, Gal Kaplun, Dimitris Kalimeris, Tristan Yang, Benjamin L. Edelman, Fred Zhang, and Boaz Barak. Sgd on neural networks learns functions of increasing complexity. An vances in Neural Information Processing Systems (NeurIPS), volume 32, pp. 3491-3501, 2019.
URL https: WtRLhhhtt::///xiiii /ttp:///xxxvvvo
abs/ /tass/995.. 1905. 11604
Ryan O'Donnell.
Analysis of Boolean Functions.
Cambridge doi: 10. 10.101770880978813994477.7..
University Press, URL https:
2014. ISBN
9781 107033333.
7/ana ysi y lan unnc .ams.org//
12

---

Published as a conference paper at ICLR 2026
Garrick Orchard, Ajinkya Jayawant, Gregory K Cohen, and Nitish Thakor. Converting static image datasets to spiking neuromorphic datasets using saccades. Frontiers in neuroscience, 9:437, 2015.
Garrick Orchard, E. Paxon Frady, Daniel Ben Dayan Rubin, Sophia Sanborn, Sumit Bam Shrestha, Friedrich T. Sommer, and Mike Davies. Efficient neuromorphic signal processing with loihi 2. In 2021 IEEE Workshop on Signal Processing Systems (SiPS), pp. 254-259, 2021.
Afshin Rahnama, Trung Nguyen, Edward Raff, Charles Nicholas, and Mark McLean. Robust design of deep neural networks against adversarial attacks based on Iyapunov theory. arXiv preprint arXiv:1911.04665 11.46666, 2019.
Nitin Rathi and Kaushik Roy. Diet-snn: A low-latency spiking neural network with direct input encoding and leakage and threshold optimization. IEEE Transactions on Neural Networks and Learning Systems, 34(6):3174-3188 2023.
Kaushik Roy, Akhilesh Jaiswal, and Priyadarshini Panda. Towards spike-based machine intelligence with neuromorphic computing. Nature, 575(7784):077-617, 2019.
Bodo Rueckauer, lulia- Alexanraa Lungu, Yuhuang Hu, Michael Pfeiffer, and Shih-Chii Liu. Con- version of aatinuuus-valluse deep networks efficient event-driven networks for image classifi- cation. Frontiers in Neuroscience, 11, 2017.
Harshay Shah, Kaustav Tamuly, Aditi Raghunathan, Prateek Jain, and Praneeth Ne- trapalli. The pitfalls of simplicity bias in neural networks. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin (eds.), Advances in Neural In- formation Processing Systems, volume 33, pp. 9573-9585. Curran Associates, Inc., cappperrrrcppppr..r..... files/paper/20207
2020. URL
File/
Neil C. Thompson, Kristjan Greenewald, Keeheon Lee, and Gabriel F. Manso. Deep learning's 's diminishing returns: The cost of improvement is becoming unsustainable. IEEE Spectrum, 58 (10):50-55, 2021.
Guillermo Valle-Pérez, Chico Q. Camargo, and Ard A. Louis. Deep learning generalizes because the parameter-function map is biased towards simple functions. In International Conference on Learning Representations (ICLR), 2019. URL https: /ttp:///ppeeeeiiieeee net net/forum?ic Syx1w23C-
13

---

Published as a conference paper at ICLR 2026
A KNOWN RESULTS IN STATISTICS
In this appendix, we recall some standard results from probability and statistics that are used throughout proof'
Lemma (Linear Combination of Gaussians). Let X....., Xn be independent Gaussian random variables with X; ~ (Ju, of), and let 01,..... On E IR. Then the linear combination
a;X;
also Gaussian random variable with mean and variance given by:
B| Σ aspiss
Var(Y) = -a.o
==1
Theorem (Chernoof Bound). Let {X; a sequence of independent random variables such that X; 1 with probability pi and X; = 0 with probability 1 Pi. Let us consider X -imm X;.
Then = E[X] = Ex=1 Ps and for all E > 0:
IPX (1 (1++))) S Se शोEP Definition 5. Let X and Y be two random variables such that
P(X > x) S P(Y > > for all x E 20) then X is said to be smaller than Y in the usual stochastic order (denoted by X <st Y
We have the following result for the stochastic domination in Binomial variables. We include its proof for completeness.
Lemma 2. If X Bin(n. p) and Y Bin(n, 9) for some 0 < <p<pp q< < 1. Then, for all l E m0, we have that X <a Y.
Proof. Let recall that the probability mass functions:
fx(k)=
12
pt
p) fx(k)
11 k
9²t (1 n)
k
and, hence, we have
(k)
(bk).
fx(k)
p
Let h(k) fx(k) (k). Since fx and fy are both probability mass functions, it holds
En h(k) = Ex (kx - (k)) = 0 and, hence, h is signed measure with total mass zero.
Since the increasing in k, (0)) < and ((n) > 1, by continuity of + there exists a (unique) 1e E (0,n) n) such that:
1. (KK) for all &s k0,
2. (k)= for all & (k*,
Therefore, we have that:
h(kk) for le (0,k") and h(k)) < 0 for k: E (k* .n]
(5)
Combining equation and Σ h(kk) 0, we notice that Σ h(j) 2 0 for all k E [n]. There- fore, for all & [0, n). obtain that
P(X k) = Σ fx(i) = 1 E Σ 1x00--r(0 =1 Σ fr Σ hG)
1=
(v) k)
which concludes the proof.
14

---

Published as a conference paper at ICLR 2026
The following result may already be established; however, since were unable to find reference, we provide a proof here.
Lemma 3. Let E (0, a0, > and X,
a,
2, tid. N(O, 1), it holds
P[X S a, px + / p2z b/ - V1 p²
|a
1
p
+
(b i
p
p
P > a, px p²z b) E V p²
+ |a
1
p
+
(b 0)
P
P
Proof. Let (p(x) == Le= x² sponding CDF, then
be the density of a normal random variable and ((r) the corre-
v2n
P > a px + p. S1]
b
pe
((x))de
Ja
p²
VI
VI pbn--b b
1
p²
4(1))
dn
p
V1- p2n
v1 p²
Vi-p"
pm
V1-
p²y b
v1 x² du
$
(())
dn
+
D(1))p
The first term can be bounded by
L
1 p²1 b
v1 1 22
00
1
(V1-
1 p²
(n)sp
dm
2
dnj
P
2
VI
p²
2 1aa 10 2/1 F 12?
e
20a
dn
V2Tp
VI 0²
2/11222
d1)
22mp
VI- Pz
pp'b²
0
in
/ p 2)
2
dn
V2mp
S SV 2
We control now the second term, obtaining that
Vi-p"
p1
VI 2 du
{())
pa
V
pz
<
(b a)
1 P
a
1+ P
v1 p
(1 p) )²)
(b a)
v 1 p
< |a|
V(1 p) p (b (b-a)
VI p2
P +
< 1a|
Combining the previous equations, we conclude the proof by noticing that
P > a, pX + =
-pX
/1 P1- > -b]
and X and -x V1 1²7 are p-correlated.
15

---

Published as a conference paper at ICLR 2026
B TECHNICAL LEMMAS
We now provide a more explicit expression for the output of sLIF neuron with threshold at time but it clearly generalizes
We give the result with sum of the input sequence, instead of the sum itself.
[0,1] by considering a weighted
Lemma 4. Let 0 0, N+, and w RR. Let consider a (discrete) sLIF newron, according to equation 7 with parameters w, 0, 'I' and B = 1, and input signal (Tt) Then, for all t E [7]. the output of the sLIF neuron at time can be computed recursively described below
8t sign (w (Ei xx) $ + +1)
80
Proof. For k E [T], we notice that
uk-1 = w
Sk-1 + +1)
then, summing over [t]. we get
Me w
8 + +)
1=1
from which, together with equation
I the result follows.
In the next lemma, we address key challenge for dynamic inputs: although each element of an input sequence 31,..., Zr lies the hypercube 111, their partial sums- procesedesssed by the neuron at each time step (cf. Lemma do not. This technical issue is absent in the static case, where the lemma unnecessary.
Lemma 5. Let 2 1, 21....,, PT 1-1,1]" V1...., VT E {-1,1]" and define h, dæ (Iz, ) h= 1 T ΣT h4, Fr = 1 ET It and VT == 1 ET Vt. Let us assume that h O(yn), then, T
T
holds that either || = | = w Tog or Jrr = | O log
Proof. Let I (i [m]
F # De,i) and I U Ie We notice that h, [T] since
Lt. 1/1 E 1-,,1)" and therefore
|
I Σ
h4 hT.
(6)
We observe that
i of = vf by definition of T;
(ii) | |ل
|
and |
|
since In 1 0.
Let us assume that ||rr||
Vn log z
Then, we have two possibilities:
1. If loo || = w
n logn
then |
lif = w
v log
and, using (ii), we conclude that
| = w
vn Tog
2. If ||
combining equation 6 and the fact [T/. S for e I and V
log n
for E 70 we obtain that
KT
> /
1 w
72
log"
We note that, since h OVyn), this scenario cannot occur for large enough.
16

---

Published as a conference paper at ICLR 2026
Finally, we conclude that either | || =
|
w
log
vn
or
Tri
O
log
v
We now give a straightforward generalization of O'Donnell spectrum concentration (Definition 4).
2014 Prop.3.3.) to the case of expected
Lemma 6. For a parametric family { probability meusure μr on W, and
& ~ Rad(1 - v). Then, for (0, 1/2].
E
Σ
(a(
S EENS, (1om
SCIn 1S|>1/1
Proof. Following
O'Donnell 2014), for fixed fwr
(a( = PPS-S [||] > 1/v],
SCIn
1//1
where the Sy, denotes the probability distribution over the subsets of [n]. which assigns probability f (S) to the subset S (recall that the Fourier coefficients to 1). Then, using O'Donnell 2014 Thm.2.49)
(fw) ESSSs 1 (1 (1-22)1111
1 (12))// IPSSs, [|5] > 1/v]
IV
(S)..
2
SCIM
18/11/2
In the first inequality, the fact that 1 - (1 20)* is non-decreasing in & is used. Then the result follows by taking expectation with respect to μ.
C PROOFS OF THE MAIN RESULTS
C.I PROOF OF THEOREM
Consider 21,.... FT E 1-1,1)" and yı. yr 111 with dy (x yt) = he for t E [T] and define 14 & Et- hr. Notice that li, [nnn] and he = [v.n]. We proceed by induction over e [T].
The base case t. = 1 requires to control P (Z1, w) + 31 (v1, w)]. We note that h h1- Let us define X Vnw" 12 and Y Vnw" Mill then X and Y p-correlated with
n
p
Vnew
|
||
Viwe
91
|
81
1=1
|| |xr]
|311
fa
2
|8/1
||,c)]
112
+
11
|
111
hn
2
|1l
17

---

Published as a conference paper at ICLR 2026
Hence, it holds that
P P[s1(11,0) w) + 81 (31, w)
= P sign((
w 21 - 0) * sign(w (w y1 0]]
n2
0 n
P X
Y - 0 1 Il_
+ P
X
S
0
Y
>
(8)
x
yy
Using Lemma
5 we can distinguish two cases
1. In the case ||r1||
1/ | = O v
we get
log
P
X
>
VI
F1
Y
S 0
1 ||
72
I IP
X
VT
[1
(1) S
E
- -8a
= O 10000"lg" (9)
n
n
n
8² 1nnl O -2 log" m)
P
X
0
Y
0
S IP
Y
0
M
1
where in (i) and (ii) we used Gaussian tail bounds. Combining equation and equation it holds
IP $1 (II, w) + $1 w)] = =O( 8 I) n) v 2. In the case |11|| |1 wi Jog
combining equation and equation & it holds
[1 w)
+ $1 $1 w]]
n
Vn
vn
Vn |
P
X
Y <
0
+ P
X <
Y
2
|/1
|
(88)
vn
1
|llya ||
|
2/1 p² 29
+
20
|>1||
P
||
Vni
1
11
||=|||||
2 PV 1 (i0)
20
20
(10)
|x1 ||
P
||0 |11l/11 lp
h1
V
hn
Vnyhu T
+ 20
+ 40
|
|11lll
| ||
21
|1
- h2
2 2
|1/1
h1
h1 log"
h1 log"
h1 log²
SCV
+ C0
+ 40C
n
n
hi log² n
$ (C(1+50))
n
where C > 0 denotes a positive absolute constant independent by the model but dependent on the data, in (iii) we have used Lemma and in (iv) we applied the inverse triangular inequality and the fact that 0 S p S 1 since h, = O log"
Combining equation and equation
10 conclude the proof of the base case, that is
h log
PP 81 (Ii, w) * si()i,
s CC
for n large enough such that
hi log" n
8 log' n
Let us now inductively assume that, for all <
1, the following probability estimate holds
The log
P(s((() + sx(1) < C(1 + 50) tV
(11)
n
18

---

Published as a conference paper at ICLR 2026
Then, have (() * ST sy)]
T-1
T-1
T-1
P
ST
# (T(),
Σ
(x(2)) +
Σ (t(
+
(T(x) + sT(Y),
8x(2t)
= Σ
sc(v)
1=1
1=1
1=1
T-1
T-1
T-1
P
8o (x( +
Σ 8e(1)
+ P
ST (T) + ST()),
Σ
8x(x)) Σ se(v))
1=1
P (s) # sc(rl IP
(T(X) ST (y), Σ
(t(2)
Σ
st(v)
(12)
C(1 + 50) VV hh
$r(x) A sr(y),
Σ
s((x)
s((1)
where in (v) we have used equation
It remains now to bound
P PIST(E1, (. .T) + sT(Y1........ Vr), ST 10-11(11111 (TI....,
T
8T( IT) # ST yT), 3r .T = 3r (.. T
First, let us define ZT = Et= and DT + Et Vr- Notice that 1 - Dr S Tor, Now, we have
t } T
(x
[11,
TT) + ST sT(11..... ,r), 32-1(21.. (8-1(21...,,T IT-1)
3r
w TT -TZ - + sign
1 TTT M/ -
sign
w
w FT
t T
T
"r
t
U w "i
t
T
Vr
t
T
T
T
ViewT T
FT
vrl
Vnww T
Ur
Vnl
t
(13)
|T>||
T|
Trrll
T
/nnw"
FT
/no
t
Vnw
T
UT
Vne
|
T|
||
Mr
Tlrr
Define correlation is
View
T
Zr |
and
Viw
T
Vr
and note that both are standard Gaussians. Their
UT
P
|| 11
|
|1
VT
i -
Vr 11
(14)
2
1 ||
[r
| || |u ||
|
17r
For
|
|
|| 11 |
ToT
2
|
|
Using equation
13 get that
IP
ST (T
IT) +
ST (V1
yr). ST ........)......
1 (J1,,..,,TT-1)) = 88 8T-1(11, v)-
xT
7
(15)
t
PP
x
70 T|
Y 5
v 70
+ PP
X < vrll
Y >
Vrll
T|
T ア
T VT
Using Lemma
distinguish two cases:
19

---

Published as a conference paper at ICLR 2026
1. In the case
Drrll =
| ||
Ol v
log ) we gete
t
P
X
Y S nn
< IP
X
v
n
(16)
T TT
Turl
T TT
of,2 log" nt 7"
S
22 1
= 0
P
n
Y > viA
I IP
y
n8
T T
T
(()
(2
|"
O
(
م2,2
log"
where in (i) and (ii) we used Gaussian tail bounds. Combining equation 15 and equation T6 it holds
(T(21, # (T......., ,UT, 8T 8T-1 (21,...., , = 8T 1111 -9T-1) T
= =0 22/2 lo
2. In the case |Brll = || || = w( log
combining equation 14 and equation
15 it holds
IP
ST
+ 8I (yi...., yr)), 88 11-11011......
1) = 88 -1(1......... vr-1) =
T
t
< IP
X > vnl
Y
S nn0
+ IP
X S Vno
y /ni
T
FT
T
T
T T|xr1
T VU
(si))
2/ 1 0²
2
Ot
VT
P-p
+ 2 0t
Vn
|
| 111
T
|T
P
T
|
|||iTT|e
(10)
01
V
01
Vig
2 1+ PV
p
2
+
4
Vn
(17)
T
P
T
||
ht
VT
hT
0t
vn
Viv hr
21
+ 20
+ 4
| ||
|
2|
||
-or
T
|
2 2
|r
< VV
hT log" n
Ot
hor log"
n
4
0t T
VV
hr log" n
T
0t)
Fr log" n
< C
1 + 5
T
n
where in (iii) we have used Lemma 3 and in (iv) we applied the inverse triangular inequality.
Combining equation 15 equation
16 and equation 17 we conclude that
P PiST (GT(x))))) [1 IT) 4 ST ( VT), अт (I),..... 2T 1) = 3 (31, VTT
T
P
(T(I (T)
IT) # sT ..... WT), 37 (T),...,TTT = 37 (2)1 -0 T
0t) T
Fy log2 n
C
1 5
1=0
hT log 2
< CT((1 50))
71
20

---

Published as a conference paper at ICLR 2026
Finally, using equation 12 conclude that
hT
T-1 ((x)
(())(2) + sT(U)
C(1 + 58){
$1
P
ST(x) # sT(().
Σ 86
Σ t=1
sc(8))
n
1=1
hT log"
$ C(1
T Σ9
n
hr log"
S cC 0)72
The proof concludes by noticing that by |rr. For static inputs, the same argument applies, but notice that there the use of Lemma | is no longer necessary. since # D2 2t J, which implies that T, and, since (-1,1)" we have ||]] /n (the same is true for y E (1),
D EXTENSIONS OF THEOREM 1
In this section, we sketch how to extend Theorem T to the case of leakage parameter / = 1, and when sign activation functions of equation T are replaced with Heaviside.
D.I CASE +
We outline below how the proof of Theorem [ extends to the general case B * 1. The overall strategy follows the proof above, but worsening the dependence on T. We believe this degradation an artifact of the proof rather than a true limitation of the model. In our main results T is treated as constant relative to n, and under this regime the conclusions remain unchanged.
Base case. The case = 1 unchanged, since does not depend on B.
2. Spike condition at time T. For > 1, the neuron spikes at time for input æ = (xt) if an event of the form
T-1
T-1 E
W
Σ
(s
20
holds, with an analogous condition for y. This changes the structure of the disagreement event (x) * sT()).
3. Key partitioning step. The analogue of equation 12 becomes
T
T-1
P(sr(z) # sT(')) P (T(Z) # ST
3
(z) +
Σ 3 3 s(())
7-1
T-1
+ P (T(2)) + (y(),,
BT- 1 (t =
ss (v)
(18)
Unlike the case 3 = 1, the term Σ1-1 1- (x)) can take 27 distinct values, which is the main source of the suboptimal dependence on T. 4. Averaged inputs. Similarly, we define
z
1
V
3
3
Vb
and
Vnw w
T
FT
Yd
vn n
V
|
1
21

---

Published as a conference paper at ICLR 2026
Similar equation 14 X and Y standard Gaussians with correlation
2
|0
11
By triangle inequality,
T
3
2
|
VT
1?
max heg- =E[T]
T
5. Conditioning on spike-history patterns. For each pattern a - (a))E//-11 E {0, 10,1)
set
As in equation 13 we must bound events of the form
==
X
n
y
n TT
plus the symmetric case.
6. Two cases, as in the original proof. We use the same case distinction from Theorem | since the conditions for Lemma still hold for o and oz.
If | = | = O(v n/ log n). then
P|E] < e aa log?
If || = l
w(V n). the same argument works after replacing t/T with gr max, ht-
di and h with E
T
Summing over all 21 patterns gives
m he log n
(P(sr()) + sT() & C2T BV
7. Conclusion. Since our main results assume is constant relative to n, the conclusions remain the same. Improving the dependence on T would require sharper analogue of equation [5] which appears challenging due the temporal correlations induced by the shared weights.
D.2 HEAVISIDE ACTIVATIONS
The extension to Heaviside activations requires tracking the norms the input vectors carefully, but the core argument unchanged. For clarity, outline the argument for the simpler case 1; for t > 1, the same reasoning applies exactly as in the proof of Theorem
1. Setup. Let y E {0, 1)" with Hamming distance = dy (x, O(vT). Define
== x,
== y,
and consider the disagreement event
03 U (X < <xX=0 0}.
2. Correlation structure. Partition the coordinates as
- {i: x = y/ = 1},
J1 = (i: 2; = 1, y/ = 0}.
J2 [t: 1 0. =
Js = {i H; 8: 0).
22

---

Published as a conference paper at ICLR 2026
Then h |J1l [J2], and
X N -
Y N
The normalized variables are standard Gaussians with correlation
1
J2
Hence, we need to bound
vn
Vn
<<0 Vn
p p:=
X
JI
v |il
V
J22
VM + [J2]
3. Case 1: = [nn). Since |]] + J2] = h = O(yn), we have 11l.1121 = O(vn). Thus,
8² n
- X 0 v exp
1 [++1)11
2
which decays exponentially, mirroring Case in the proof of Theorem
4. Case 2: | wn).. this regime, p is close 1. Applying Lemma yields
1- P + 0 Vnl |11 J2
ps v
J1
P
VIT
+ J1 7 J2l
which matches the scaling from our original proof. Hence the same stability bound holds for Heaviside activations.
Using |/| wn) and h = O(vn) gives o
D.3 PROOF OF THEOREM 2
Let denote Fy Ek= Thi VT Et Uk and (1 ve) [umn] with is [0, 1]- Let us define v = [T) 14 and assume v O (a We follow similar strategy to that of Jonasson et al. 2023). Notice that, for all I E [L], the probability that (1) (21,......, W) and (V, yr, W) differs depends only on the number of neurons that have at least one disagreement at any time at layer 1 - and not on where they disagree. We define DJ as the number of neurons at layer that have at least one disagreement at any time, that is D D. max (2), .......,; W) 2 (v........ yk. W)). With this, D is Markov chain with + states, where D
is an absorbing state. More precisely, let us denote with M; = 2 maxke[T] (1 (x1..,, 2k, W) sk ( Uk. W)). Notice that Mi,..., Mn are in- dependent random variables with value in {0. 1}. In particular, exploiting Theorem I it holds that
PMM 1| = P
s ( kEl27
max
(1)
W) - s 11) (or..... U/. W) 2
1T
(1-1)
log²
TT((1+ 8)
n
D (1-1) log"
n
for all i. E [n]. Therefore, 100 D. k is Binomial random variable upper bounded in the stochastic sense by
k log²
| (1-1) = ~ Bin n, n9T + +01
(19)
23

---

Published as a conference paper at ICLR 2026
thanks to Lemma 2
We invite the reader to see Definition for rigorous definition of smaller in the
usual stochastic order. We write now h4 = [uu'] for some V, E [0, 1] and define v [ Vi the maximum number of input disagreements.
We proceed by induction over the depth dimension E [L].
Let start from the base case = 1. We want to estimate
n
(D ( 2 v 1/4 n logngg = Σ
Pw ( 2 1/4 n logn
DT = k) Pw (D = k)
Pw
[um]
Pw (D4²) nlog D = k) Pw (D,-= =k) + PW (D > lmn))
Pw (Df) 2 vn
DJ pm
where in (i) we used the stochastic dominance and the monotonicity of the probability
appearing in equation 19 Now, combining Theorem 3 with & 1, which is admissible
given that ला
1-1/4
2 and v S J. for n large enough, we conclude
Pw (D (x, 2 I logn D (z. v) = (mn)) S Stvin
Let assume by induction that
Pw (D 2 Sズt nlogg) s lle -tozttt
(20)
Then, we have
Pw
D D1(42)
2 1//220443)
n mlogn
Σ PP (D 2 1/22441)
nlog D = k6) IPw (Df) = k))
k=1
(t log n
Pw (D(+t)) 2 11/22011) n logg n D = k) IPw (D = kk k))
1=1
+ Pw (D9) > u x n mlgg
(21)
v22n logn
PwW (D(ttt) 2 v n log DT = k) IPw (DI) = k) + le tuzzt
< IPw (D
vondtn n log n D | | log )) + le guzzt
q Vlog wi CCC(14)
Now, using Theorem with s = 1, which is admissible because v < v for n large enough, we conclude that
> 2, and
Pw (D (D(+)((,)))
(x,4) v) 2 ( n log Dx (x, v) | nlgn )
le cer2211t
(22)
Hence, combining equation 21 and equation 22 we conclude that
Pw (Df) (x, v) 2 waati n logn) < le -e--v I4TT IJ.FT
24

---

Published as a conference paper at ICLR 2026
Now, if L = 1, can apply directly Theorem obbtiinin that
Pw (fhT (x, W) + fl. (y. W))
T
T
1 Se ((F)) W) # arg max
1 ((ve)eer)) w)
Pw
arg max
sk ((rr))err) W) -1 ((N))))) w) w> 1>0)
< IPw
max ke(
S nn C(1 + 0) log nvv.
If L 2, using Theorem conclude that
Pw
pL. (r, W) + fL. (L.
(1) 1 ((I))E)M) W) 5/ ((oobett, W) W)) 1=0)
(1)
$ Pw
max (ET
< Pw
max (E
(D) ((I)) 1
( ((vckeeT) w) D(L-1) s luzde n log
+ (L- 1)e -0
S nr. Sn,TTC(1 0) log (L- 1)e
which concludes the proof.
D.4 PROOF OF COROLLARY
Let v, E 1-1,1) 13" be static inputs. Define N(g) Ei + 1). Then, we have for Unif([-1,,1) and & ~ (la(1-1)
ENS, (IFLIT(, (W))w-woo..))
(1,
Ex,g [P (fL.T (z. W) + (LT
Fe Pw (JZ.T(,, W) # pd. (To N(E) VT) (PE(N(() s vn)
+ Ex PW (fLiT W) # FL. (x 2 ((E) > vn) P&(N((E) > vn)
S CT.OOU
2 24++
3/2 n + (I 1)e
+ +e- iv
In the last line used Theorem with du (2, x @ E) = LVTJ. and
with v', which applies because N(E) < LVn] is equivalent Vn (by assumption). We also used that
P (NV(E)
[vnJ) < tiv tva
which follows from applying the Chernoff bound (in Theorem 3 to the random variable N({)) Bin(n, The statement about the expected degree concentration follows applying Lemma 6 using the bound on the expected noise sensitivity above.
E ADDITIONAL EXPERIMENTS
In this appendix, we report additional experiments that complement the results presented in the main text. These include further evaluations of noise sensitivity and analyses of (s)IF spiking neural networks under different settings.
Single LI) Neuron. We present additional experiments on (s) spiking neuron. These results complement the main text by providing additional empirical support for the claims made in the experiment section. In Figure 7| report results for the neuron with threshold 0 0. In this case, we observe that the neuron either fires at all times or does not fire at all. This is consistent with the experiments, as the sensitivity remains constant over time.
25

---

Published as a conference paper at ICLR 2026
sensitivity
Expected sensitivity
332
030
1'e-122
ENS
ENS
Theertoor Bound
308
348
030
Dimension
Input Dimansion (n)
(a) sIF single neuron (fixed2)
(b) single neuron (fixed2)
Expected naise sensitivity of sif with == Geo
Expected nolse sensitivity of with v
2.2
ENS
Input Dimension (n!
Input Dimension (n)
(c) IF single neuron (vn log 12)
(d) sIF single neuron (vT)
Figure 5: Noise sensitivity ENS2/,and ENS, /1/m n) for different input dimensions n for sIF and IF neurons with 0 = 0.5 and T = 10. (a) ENS for sIF neuron. (b) ENS for IF neuron; (c) ENS loggn) for sIF neuron. (d) ENS, ENS//(/RR lo for IF neuron.
26

---

Published as a conference paper at ICLR 2026
Expected noise sensitivity of #LIF with v
Expected noise sensitivity of LIF with v=
ENS.
ENS
311
0.0
308
Input Dimension (n)
Iniut Dimension Ir)
(b)
Expected noise sensitivity of SLIF with Igent
Expected noise sensitivity of UF with waard
1.50
ENS ENS
049
0.0
Ingut Dimension (n)
Input Dimension
(c)
(d)
Expected sensitivity
Expected sensitivity
67
ENSu=
ENS
01
Input Dimension
Input Dimension (n)
1/(v/ log for different input dimensions n for sLIF and LIF neurons with 0.5, T 10 and 0.5. (a) ENS for sLIF neuron. (b) ENS for LIF neuron; (c) ENS 1/(Vnloon) for sLIF neuron. (d) ENS, / / for LIF neuron.
Figure 6: Noise sensitivity 2NS2/maad and ENS
27

---

Published as a conference paper at ICLR 2026
Expected sensitivity of with v=
Expected naise sensitivity of IF with
ENS
Thoee of Board
ENS
Therator Eoun
0:3
Input Dimension (n)
Input Dimension (n)
Expected naise sensitivity of siF with
Expected noise sensitivity of IF with
8.30
033
2.2
023
0.33
ENS
0.33
0Pe-10
0:3
-1
8:35
045
6:30
040
Input Dimension (n)
Input Dimansion (n)
(c)
(d)
Expected noise sensitivity of sIF with v 2
Expected noise sensitivity of IF with v
Iogaon
022
ENS
ENS
0.35
Dood
Input Dimension
1
Input Dimension
(e)
Figure/ 7: Noise sensitivity ENS. and ENS, ///ll///oo for different input dimensions for sIF
and IF neurons with = and T' = 10. (a) ENS for sIF neuron. (b) ENS for IF neuron; (c) ENS /t log for sIF neuron. (d) ENS 1/(a l) for IF neuron; (e) ENS 1/(5) for sIF neuron. (d) ENS 1/(/n) for IF neuron.
28

---

Published as a conference paper at ICLR 2026
Deep (s)IF SNN. We present additional experiments deep (s)IF spiking neural networks. These results complement the main text by extending the analysis of noise sensitivity to multi-layer architectures.
Expected sensity of S-layers siF SNNs at v=
Expected noise sensity of 5-layers IF SNNs a
0.26
ENS
0.24
0.22
ENSus2
0.12
0.20
ENS
0.38
0.10
0.16
8.14
Input dimension (n)
Input dimension (n)
(a) sIF single neuron (fixed2)
(b) IF single neuron (fixed2)
Expected nolse sensity of S-layers IF SNNs at v
Expected noise sensity of 5-layers siF SNNs at
1MA7. + ENS
0.35
0.38
0.34
EWS
0.11
0.12
ENS
0.37
0.10
0.30
0.09
0.8
Input dimension (n)
Input dimension (n)
(c) IF single neuron (v log n)
(d) sIF single neuron (v
Figure 8: Noise sensitivity ENS v for different input dimensions for 5-layers sIF and IF neural networks with 0 = 0.5 and T 10. (a) ENS2/" for 5-layers sIF neural network (log-scale x-axis); (b) ENS
1/ n) for 5-layers sIF neural network (log-scale x-axis).
29


Reviews:
[{"review_id": "Z2FpO6bvBZ", "reviewer": "Reviewer_77v2", "summary": "This paper investigates the theoretical stability properties of discrete-time Leaky Integrate-and-Fire (LIF) Spiking Neural Networks (SNNs). The authors employ Boolean function analysis to quantify noise sensitivity and connect it to a new notion of spectral simplicity, showing that wide, randomly initialized SNN classifiers are on average stable and biased toward low-frequency (simple) functions. They provide analytical bounds for both single-neuron and multi-layer cases (Theorems 1–2), establish a formal relationship between stability and Fourier spectrum concentration (Corollary 1), and validate the theoretical insights through small-scale simulations on MNIST and CIFAR-10.", "questions": "1. Could the author add any real world event benchmark like DVS-CIFAR10, SHD or larger networks to strengthen the empirical section. \n\n2. Could authors add an example showing the power spectrum to substantiate the “spectrally simple” claim? \n\n3. Would be nice to compute for tiny inputs, the cumulative spectral energy vs degree to visually confirm low-frequency dominance.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Applying Boolean function tools to spiking dynamics is original and mathematically elegant. It connects neuromorphic computation with the mature literature on noise sensitivity and Fourier analysis. The paper is clear, with rigorous definitions (noise sensitivity, spectral concentration) and clean proofs for Theorems 1–2", "weaknesses": "1. The paper shows minimal experiments and primarily confirm qualitative trends. The paper would benefit from convincing demonstration of the claimed real world persistence of spectral simplicity in trained SNNs. \n\n2. The experiments measure noise sensitivity directly but there are no empirical validation of spectral concentration using Fourier spectrum.\nto verify spectral concentration empirically. \n\n3. Can we also claim that spectral stability will imply parameter stability? For example, may be authors could try injecting Gaussian noise into weights or thresholds and measure output sensitivity with the model. \n\n4. It would be nice to add evaluation for the output with randomly dropping 5 % of spikes or shortening the readout horizon TTT. C", "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": "jO2odJ0Yqc", "reviewer": "Reviewer_rx3u", "summary": "This paper analyzes the stability of randomly initialized spiking neural networks (SNNs) with sign integrate-and-fire (sIF) neurons, providing stability boundaries for both individual sIF neurons and multilayer sIF networks. Theoretical analysis proves the stability of random SNNs. This paper also performs spectral analysis, demonstrating that randomly initialized SNNs tend to exhibit simple spectral structures with low-frequency components.", "questions": "1. Please provide the stability bounds for a general $\\beta$.\n2. Please provide the stability bounds for the typical LIF model.\n3. This paper employs Xavier initialization with variance set to 1/n. However, this initialization method is not specifically designed for SNNs and is unsuitable for activation values with non-zero means. I wonder how alternative initialization methods, such as Kaiming initialization or initialization methods specifically designed for SNNs like [1], would affect the stability bounds.\n\n[1] Ding, Jianhao, et al. \"Assisting Training of Deep Spiking Neural Networks With Parameter Initialization.\" *IEEE Transactions on Neural Networks and Learning Systems* (2025).", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper investigates the stability of SNNs from the perspective of Boolean function analysis, offering a novel viewpoint for SNN stability research.\n2. This paper presents a detailed theoretical analysis and derivation, demonstrating a solid technical foundation.", "weaknesses": "1. This paper assumes that the leakage parameter (membrane time constant) of LIF neurons $\\beta=1$, thereby reducing LIF neurons to IF neurons. Compared to simple IF neurons, LIF neurons are more commonly used and exhibit more complex neuronal dynamics. This paper does not provide further analysis to demonstrate how $\\beta$ affects the stability bounds.\n2. This paper employs sign leaky integrate-and-fire (sLIF) neurons. This neural model extends Boolean functions to the discrete-time domain. The sLIF neuron produces values of -1/1, which differs significantly from the typical LIF neuron that fires 0/1 spikes. This paper does not provide stability bounds for the typical LIF model, which holds greater significance for SNN stability research.", "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": "sIccwRWdBp", "reviewer": "Reviewer_jtsR", "summary": "This paper focuses on noise sensitivity and quantifies the bound on the stability of SNNs. A notion of spectral simplicity is introduced, motivated by this bound. Several experiments are conducted to verify the proposed method.", "questions": "1. Figure 3. onMNIST should be \"on MNIST\"\n2. Could you further explain what the conclusion is for the expected noise sensitivity data shown in Figure 1? Does this Figure just show sIF and IF neuron has similar noise sensitivity? How are the contents related to the summary of Section 5 on line 375 to line 377?\n3. \"Notice that training reduces the model’s sensitivity,\nbut less strongly compared to MNIST. This aligns with the fact that both the training and test errors\nare larger for CIFAR-10 (which achieves 84.38% training accuracy).\nWhat are the training and test errors described here?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The abstract is clear and well written.\n2. The results shown in Figure 3 are interesting.", "weaknesses": "1. Experiments are very simplified and the scope is limited\n2. The format of the paper can be further improved. Section 1.2 (Notion) in the Introduction section may not be appropriate, and it may be better to put it in Section 2. The bolded \"Future directions\" in the Conclusion section is confusing. Is it a section/subsection title? Why is a term bolded and put there? Suggesting to revise it to make the format coherent in the paper.\n3. The conclusion does not summarize the results from numerical experiments. It will be beneficial to add them here to cover all key contents of the paper.", "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": "MQD3Ln7xFE", "reviewer": "Reviewer_7NY3", "summary": "The paper studies stability of discrete-time leaky integrate-and-fire (LIF) spiking neural networks (SNNs) through Boolean function analysis. It proves that wide, randomly initialized LIF-SNN classifiers are noise-stable on average: small random input bit flips are unlikely to change the predicted class. From this, the authors introduce spectral simplicity—a notion saying most Fourier–Walsh mass concentrates on low degrees—and show random LIF-SNNs are biased toward such “simple” functions. Experiments on single neurons, deep networks, and trained models (MNIST/CIFAR-10) suggest (i) both IF and “signed” IF variants are noise-stable, (ii) depth increases sensitivity but less than the theory’s worst-case bounds, and (iii) training further reduces sensitivity.", "questions": "Can parts of the analysis (Theorem 1/2) be ported to the classical Heaviside LIF (without sign) to close the theory–practice gap you highlight empirically? Which steps fail, and can they be patched (e.g., via comparison lemmas)?\n\nYour proofs suggest stability worsens with L and T, yet experiments look kinder. Can you isolate which proof steps (union bounds, reset dependencies) are driving pessimism and provide refined bounds (e.g., martingale or coupling arguments) that track empirical trends?\n\nHow sensitive are your results to input distributions? Could the corollary be re-stated under a product measure with bounded correlations, or for common dataset distributions after whitening? You mention generalization is “straightforward”—can you sketch it?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Stability/robustness theory for spiking models is far less developed than for ANNs; bringing Boolean analysis + Fourier tools here is novel and interesting.\n\nSpectral simplicity offers a principled bridge from stability to a simplicity notion compatible with SNNs and complements prior simplicity-bias results in ANNs. \n\nThe signed LIF formulation and spike-count readout are standard enough to be meaningful while still analyzable; proofs acknowledge the reset-by-subtraction complication. Also, if random LIF-SNNs are spectrally simple, that has consequences for generalization priors and for neuromorphic design choices (thresholds, depth, latency).", "weaknesses": "The strongest theory is for random weights and static encodings; dynamic inputs are acknowledged but technically harder (partial sums leave the hypercube). This narrows practical reach (e.g., event streams).\n\nThe corollary infers low-degree concentration from ENS bounds, but there’s no empirical Fourier probe of trained models to confirm the spectrum actually concentrates as predicted.", "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": "kFtYHUEXk1", "reviewer": "Reviewer_KsVi", "summary": "The paper builds on previous theoretical work and investigates the stability of random Leaky integrate-and-fire spiking neural network classifiers. The question is, how is the output of the network sensitive to the changes in input and when the changes in the input change the classification. While the majority of the work is theoretical, a minor part consists of numerical experiments that test the validity of theoretically computed bounds on training hyperparameters. It is found that random spiking networks are stable, in the sense that small changes to the inputs do not change the classification outcomes.", "questions": "1) [line 162] Authors give a definition of what they call a recurrent network, but to me it seems that they define a multilayer feedforward spiking network - a network where each neuron in layer $l$ receives spiking output from neurons in the previous layer $l-1$. In what sense is this a recurrent network?\n\n2) Proof sketch : step 3 reads somehow unclear, can the authors explain better what is done in the last step?\n\n3) Why does Figure 2 not show the bounds?\n\n4) Figure 3 shows the sensitivity as a function of \\nu. Why is that?\n\n5) The dependance of the result on log n in Theorem 1 is not clear to me. Can authors explain?\n\n6) Authors perform their analysis on random spiking networks. Would results apply to spiking networks with structured connectivity (e.g. Koren and Panzeri, NeurIPS 2022; Urdu et al., ICANN 2025)? If not, what is fundamentally different in structured vs random networks that would prevent such generalisation of your results to structured networks? If authors find the question relevant, I suggest discussing it in the revision.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "While the paper builds on previous results, it also brings original contribution to previous works. The paper is logically constructed, motivates well the interest of the topic and brings significant results on the stability of SNN classifiers. The paper is rigorous and brings a valid theoretical advancement to the understanding of spring neural networks. Rigour and clarity are, among others, achieved through defining and explaining all quantities used in equations.", "weaknesses": "A sizeable portion of the paper presents previous results. This is in itself not a problem, as it serves the purpose of presenting a complete theory and because new results would be hard to put into context otherwise. However, on several occasions it remains to some degree unclear what is a previous result  and what is a new contribution. Authors could be more specific about this as they unfold their results. \nMoreover, some results remain unclear and could be better explained. In some places, the presentation would profit from clarification, better intuitive explanations and on further comments of obtained results.  More commentary and intuitive explanation could be provided along the Results part of the paper to allow to the reader to better understand and appreciate the significance of the results. Finally, the paper does not have a significant limitation section, which would be important to have.\n\nMinor:\nThe y-axis of Figures 1 and 2 is \"ENS\" and the y-axis of Figure 3 is \"noise sensitivity\".\nx-axis of Figures 1 and 2 report the number of features, but in the text they are referred to as the \"input dimensions\".  It is advisable to unify the axis notation.", "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)