BioOSS: A Bio-Inspired Oscillatory State System with Spatio-Temporal Dynamics
Authors: Zhongju Yuan
Geraint A. Wiggins
Dick B.M. Botteldooren
Venue: NeurIPS 2025 poster ()
Keywords: oscillatory state system
spatio-temporal dynamics
bio-inspired model
time series tasks
Primary area: neuroscience_and_cognitive_science

Abstract: Today’s deep learning architectures are primarily based on perceptron models, which do not capture the oscillatory dynamics characteristic of biological neurons. Although oscillatory systems have recently gained attention for their closer resemblance to neural behavior, they still fall short of modeling the intricate spatio-temporal interactions observed in natural neural circuits. In this paper, we propose a bio-inspired oscillatory state system (BioOSS) designed to emulate the wave-like propagation dynamics critical to neural processing, particularly in the prefrontal cortex (PFC), where complex activity patterns emerge. BioOSS comprises two interacting populations of neurons: p neurons, which represent simplified membrane-potential-like units inspired by pyramidal cells in cortical columns, and o neurons, which govern propagation velocities and modulate the lateral spread of activity. Through local interactions, these neurons produce wave-like propagation patterns. The model incorporates trainable parameters for damping and propagation speed, enabling flexible adaptation to task-specific spatio-temporal structures. We evaluate BioOSS on both synthetic and real-world tasks, demonstrating superior performance and enhanced interpretability compared to alternative architectures.

BioOSS: A Bio-Insppired Oscillatory State System with Spatio-Temporal Dynamics
Zhongju Yuan Geraint Wiggins Dick Botteldooren'
WAVES Research Group, Ghent University, Gent, Belgium
2 Al Lab, Vrije Universiteit Brussel, Brussel, Belgium EECS, Queen Mary University of London, London, UK
zhongju. yuan@ugent. be, geraint.wigginseee wiggins@vub. be, dick. bickk.bottelldooreddneentt be
Abstract
Today's deep learning architectures primarily based on perceptron models, which do not capture the oscillatory dynamics characteristic biological neural activity. Although oscillatory systems have recently gained attention for their closer resemblance to neural behavior, they often lack structured mechanism to represent rich spatio-temporal dynamics in controllable and interpretable manner. In this paper, we propose a bio-inspired oscillatory state system (BioOSS), a 2D to- pographically organized oscillatory state-space model designed to generate diverse oscillation-drrvenn spatio-temporal patterns. BioOSS comprises two coupled state components: p units that represent pyramidal-ced activity, and units that act as velocity-like latent states controlling phase, time scales, and damping. The model incorporates trainable parameters for damping and effective oscillation rates, enabling flexible adaptation to task-specific temporal structures while remaining efficient for long-sequence learning via scan- friendly diagonal dynamics. We evaluate BioOSS on both synthetic and real-world tasks, demonstrating superior performance and enhanced interpretability compared to alternative architectures.
catorrnn--ooentialliiike variables inspired by
Introduction
The advent of deep learning models, such as transformers (Vaswani et al. 2017 Devlin et al. 2019]] has sigmificantly advanced performance across variety tasks and domains. However, these models are still fundamentally based on perceptrons, which differ from the oscillatory behavior widely observed neural activity. contrast, oscillatory systems (Rusch and Mishra 2021 Lanthaler et al. 2023] Rusch and Rus 2025) have recently garnered considerable attention due to their closer alignment with dynamical properties of biological circuits. While these models capture temporally recurrent and oscillation-like behaviors, they often provide limited control over how spatio-temporal structure is organized and interpreted. In biological circuits, oscillatory activity emerges within structured substrates (e.g., cortical columns and topographic organization) and can manifest coordi- nated, wave-like activity patterns that appear to spread across tissue. Such phenomena are shaped by heterogeneous time scales and attenuation, and by the circuit's structured layout. Consequently, exist- ing oscillatory models primarily emphasize temporal dynamics in an abstract latent space; even when spatial extensions considered, they often do not explicitly leverage a topographic organization that supports interpretable spatio-temporal pattern formation.
We propose a bio inspiedd oscillatory state system (BioOSS) to learn spatio-temporal dynamics that can exhibit wive-like coordination patterns on a structured substrate. In the brain, neural states
"Coressponiing author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Bio Framework
Output
(b) Neural Network Structure
(c) Oscillator Frequency Distribution of neurons
C
Nonlinear Layer
Grid Y
2DD Oscillatory Layer (p field)
Input
Grid X
WMMNNN WMMMMMM
MMMMN VMMV
(d) BioOSS Oscillation Pattern
(e) Linear Model Oscillation Pattern
Figure 1: (a) The overview of the proposed BioOSS framework. (b) Structure of the 2D neural network composed of coupled p wnembrrne-potential-o on a spatial grid. (c) Distribution of the natural frequencies across the grid, illustrating local frequency heterogeneity. (d) Dynamic behavior of p units in the proposed model. The left panel shows the input signal to a left p unit; the right panel shows a wave-like, coordinated spatio-temporal response pattern neighboring unit under the learned grid grid-structured dynamics. (e) Behavior in the absence of nonlinear dynamical enrichment, where the right unit exhibits only amplitude modulation without generating richer oscillatory structure.
and o (velocity-like latent) units arranged
evolve not only over time but also within spatially organized tissue, where coordinated population activity supports higher-order cogmitive functions. Such organization particularly salient in the prefrontal cortex (PFC), where complex and coordinated activity patterns support cognitive control and integration (Dale and Halgren 200T Buonomano and Maass 2009]. Temporal recurrence and delay mechanisms enable memory and integration [Fuster and Bressler 2012 Richards and |Fanklladd 2017), while cortical columnar organization provides an anatomical scaffold for structured interactions that can give rise to locally coordinated, wave-like activity (Watakabe et al.) [023]] BioOSS inspired by these observations and provides a unified, interpretable framework for learning task-relevant spatio-temporal dynamics (Fig. [b).
As observed in [Watakabe al. 2023). neural circuits tend to organize primary gegnal-carrying units into structured rows rather than random patterns. To capture this property. we propose a 2D grid-based framework composed of two coupled state components: p and units (Fig. (Fig.[ib). p units serve as the main signal carriers in our model, functionally inspired by pyramidal-cell coemiddl--ell-doneee -elel-lllmmniteedd activity within cortical columns [Watakabe al. 2023). and exhibit heterogeneous oscillation preferences across the grid (Fig. TT). In parallel, units act as velocity- -ike latent states that control phase, effective time scales, and damping, thereby shaping how local activity evolves and how coordinated, spread- like patterns emerge at the population level. The resulting grid-structured dynamics can produce wave-like spatio-temporal patterns beyond simple harmonic components (Fig. [dd). contrast, purely linear transformations, such as fully connected layers without non-linearity, only reweight and shift existing components without introducing richer dynamical structure (Fig. Ib).
To introduce biologically plausible flexibility, we incorporate trainable parameters that regulate attenu- ation and oscillation rates, including damping and a wave-inspired speed parameter c that influences the effective oscillation rate under a stable diagonal modal update. By learning these parameters from data, the model adaptively generates task-relevant spatio-temporal activity patterns while remaining efficient for long sequence learning. Importantly, BioOSS is not positioned as full stencil- based wave solver; rather, it provides scan-friendly oscillatory state-space parameterization on a 2D grid that can express coordinated, wave-like activity patterns in a controllable and interpretable manner. Furthermore, the grid structure provides an interpretable substrate for analyzing learned dynamies, including mode-based characterizations of the resulting ppatio-tempora patterns [ia et al. 2024].

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In summary, our contributions are:
We introduce BioOSS, biologically inspired, grid-structured oscillatory state-space model that can express coordinated, wave-like spatio-temporal activity patterns topographic substrate, without claiming to be a full siencil-based wave propagation solver.
We design an efficient trainable framework with flexible damping and wave-ine save-inspired rate parameters, enabling adaptive and stable long-sequence learning via scan-friendly diagonal dynamics.
We validate BioOSS through extensive experiments on synthetic and real-world datasets, demonstrating both competitive performance and enhanced interpretability compared to baseline architectures.
2
Related Work
Neuroscience Inspiration. Our model is inspired by neuroscience findings that cognitive functions and information processing are associated with coordinated oscillations across neuronal populations.
Such oscillations, shaped by the interaction between excitatory and inhibitory processes, can give rise to rhythmic and synchronized activity patterns [Doelling and Assaneo, 2021 Esghaei et al. 2022). Recent studies [Watakabe et al., [2033] suggest that neurons the prefrontal cortex (PFC) exhibit structured, mosaic-like organization, where signal-carrying units interact through intermediate circuitry and diffuse projections. This anatomical scaffold, together with heterogeneous local time scales, is believed to support diverse population-level dynamics. Building on these observations, we design a simplified 2D topographically organized system with two coupled state components. By learning per- per-location damping and oscillation-rate parameters. wave-like ppatio-tempora activity patterns and resonance- behaviors that qualitatively resemble population dynamics reported in biological circuits [Foster and Scheinost 2024 Rigotti et al. 2013).
model can produce coordinated,
Comparison Existing Sequence Modeling Approaches. Sequence modeling captures temporal structure in both biological and artificial systems. Transformers (Wen et al. 2023) model dependencies through attention, RNNs [Torres et al. [022]] maintain hidden states over time, and State Space Models (SSMs) Auger-Méthé et al. 2021] describe latent dynamics via structured recurrences.
Oscillatory SSMs explicitly parameterize rhythmic dynamics. LinOSS [Rusch and Rus 2025 derives ccan-compatible recurrences from oscillator dynamics with stable discretizations for long sequences. BioOSS is related but adopts a different inductive bias; it organizes states on 2D topographic grid with heterogeneous per-location parameters and separates gegaribbg (p) from modulatory (o) components for interpretability. In its scan cean-optimized form, BioOSS uses diagonal (or approximately diagonal) modal dynamics, describing wave- like behavior as coordinated spatio-temporal patterns rather than explicit stencil-based neighbor propagation.
Related directions include grid- based wave models with convolutional couplings (c.g.. Neural Wave
Machines [Klller and Welling 2023)) spectral graph filtering [Balkenhol et al. 2024], and analog
propagation substrates such as Wave-RNNs (Hughes et al. [001 and water-wave reservoirs [Maksy] mov 2023). These approaches share the goal of oscillation- or wave-inspired computation but differ in their coupling mechanisms and computational trade-offs.
3 Methods
3.1 Overview of the proposed model
We propose BioOSS, illustrated in Fig. [a. whose core component is two-dimensional oscillatory layer (Fig. [T). This biologically motivated model consists two neuron types: p neurons, which serve the primary computational units and carry the signal, and neurons, which modulate the effective oscillation rate and phase progression across the grid, shaping how coordinated spatio- temporal patterns emerge. The spatio-temporal dynamics of the system are described by the following
3

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equations:
Op k + V o = Bu(t).
(1)
0t
do
o 0 = 0,
(2)
01
where denotes the p neuron value, o = (0z, 02) represents the neuron values in the æ and directions, k2 and k are the damping coefficients, is the wave speed, and u(t) is the external input at time step t, while B is the input weight matrix. For convenience, the system can be expressed in matrix form as:
x1)) = - k2
V
B 0
B 0
x())
+
u(t) Ax(() +
u(t),
(3)
where x(t) = Ip. of and A is the coupling matrix.
The output equation is given by:
y()) = Cx(() + Du((),
(4)
where C and D are the linear weight matrices.
Temporal Discretization Scheme. To mitigate the significant computational cost associated with solving the PDE at each time step, we propose an explicit discretization scheme that generates an explicit state transition matrix. This approach avoids the need for a solver, enabling more efficient computation.
We first apply intermediate coupling update with fixed timestep A/ as follows:
O = On-1 AtVpn-1.
(5)
Pr Pa-1 AiVV o + A/Bu..
(6)
Next, we apply damping correction as follows:
(I Atk o, O
(7)
(1 Atk?) Pr Pn
(8)
The full update equation is then given by: X Damp
M Velocity Xn-1
+ Damp F Velocity
A Axn-1
+ Damp
F Velocity
1
where M Velocity
[ I
-
Damp
Alke
It Atke and Velocity
AAV
I
A/Bu,
Although A Damp M Velocity serves as coupling matrix to capture internal propagation.
Since Damp is diagonal by construction, its inverse is obtained by taking the reciprocal of its diagonal entries:
Damp
diag I+ At 1+ K
(10)
where each diagonal entry is simply the reciprocal of the corresponding factor (1 + At At,))) or (1 + At k22) at grid point (i.j).
Therefore, the coupling matrix becomes the following form:
(I+ A -(-AIT A
- Atke) -V. (I Ark)-
A Damp Velocity
(11)
Since and are defined on a discrete N x N grid, c, k", and k' are per-cell parameters that may vary with location (i,j). For spatially constant parameters, the periodic-grid wavenumber basis diag- onalizes the spatial operators; with spatial heterogeneity, Fourier modes generally mix. To preserve cean-friendly diagonal recurrence, we use wave-inspired per-location modal parameterization and assume and k vary weakly that cross- mode coupling can be neglected.

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3.2 Efficient Recurrent Scan Operator
The computational efficiency of BioOSS is limited because the recurrence is inherently sequential and cannot be parallelized, while the spatial operators (gradient and divergence) must be applied at every time step. To alleviate this cost, we adopt the sean operator together with an eigendecompositinn based diagonal approximation of the global state-transition operator. After spatial discretization, the resulting recurrence couples all grid variables through a large sparse but non-diagonal matrix, which prevents efficient parallel scan updates. Our approximation restores diagonal recurrence that is compatible with associative sean, reducing the cost for long sequences.
The main challenge is that the recurrence induced by these operators breaks the associativity property (x=y) -r (1)=) Rusch and Rus 2025). Here, I, y/ denote intermediate system states, typically
represented as tuples (A, M.Mpp F Velocityy combining the system matrix and the input vector at each step, and is binary operation such (a1,a2) (b,; b2) a1, b b2 by) [Rusch and Rus 2025 As emphasized in recent work on oscillatory state space models (Rusch and Rus 2025] associativity is essential because it enables parallel scan operations, reducing the computational
complexity of a serial recurrence from (N) to (log N). Following SSM literature [Orvieto et al.
2023 Rusch and Rus 2025 we therefore apply eigendecomposition
PAP where A is diagonal. This diagona lization restores associativity and allows efficient scan operations across time steps, significantly lowering the computational cost of sequential updates.
To find the eigenvalues, we analyze the periodic-grid wavenumber representation, where discrete gradient and divergence become algehraic multipliers. This is exact for constant coefficients; with spatially varying c and k, products induce convolutions & and mix modes, so we assume c and k vary slowly and neglect cross- mode coupling to first order (Appendix A). For spatial frequency
components (En, Ey) and for each grid point (i,j) with local values of Ci.j- (.5.
LP and kgz
Vp + (iExp, iLyp) V o + i5 + iEy0y
(12)
(13)
For each Fourier mode (Eu, Ev) and each local grid point, the system matrix becomes a 3 x 3 matrix:
dA THAIN
aatic T+AI 4J
THAIR AtiCE AIke
(14)
T+AI
ANEE
14
I+
On a bounded N x N grid with periodic boundary conditions, the admissible wavenumbers (Ex, Ev) are discrete. In the homogeneous (shift-invariant) case with spatially constant c, k", and ko, the global discrete operator becomes dlick-diagonal in this wavenumber basis, with one 3 x 3 block A Ey) per discrete wavenumber. The cigenvalues are therefore obtained by solving the characteristic equation det(A(Ez: En) - AI) 0 for each block. which yields cubic characteristic polynomial that can be written in the following factored form:
(1 (1+A - [1+
1 (1+ ->)
AA + +2) (1+ +IKK -))+ (1+ (1 Atke,) = (15)
1
From the factored characteristic equation, we can identify two distinct eigenvalue solutions: a real eigenvalue corresponding to one component of the velocity field, a I+ and the other two
eigenvalues are the solutions to the right part quadratic term.
Applying the quadratic formula to the factored characteristic equation, a complex-conjuee eigen- value pair arises whenever the discriminant is negative, corresponding to an oscillatory regime. On a finite periodic grid, the admissible spatial frequencies (5x, Ey) are discrete and bounded by the
grid resolution Ar and the grid size, so || v <2 + E should be interpreted as chosen from this discrete set rather than taken arbitrarily large. Under the mild approximation hr, ~ kf this yields the following convenient closed- fooeed-form expression for an oscillatory
conjugate pair:
C. At
E2 + 52
1
1 AIR A
A2,3
+ 1
Ei
(16)
1+
Alko
(1+
AIR )1 + A
5

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When k2, and k" are spatially constant, Eq. (66 matches the eigenvalues of the global shift- invariant discretized operator indexed by (Ez, Eg) In BioOSS we allow spatially varying parameters (e.g., Cij), which breaks shift invariance and generally mixes Fourier modes; thus Eq. (16) is used as wave-inspired per-location modal parameterization for initialization and scan-friendly diagonal recurrence, rather than the exact spectrum of the full heterogeneous operator.
Given the per-location diagonal modal parameterization in BioOSS, the corresponding modal basis can also be constructed in closed form. The complete eigenvector matrix is then assembled as follows (details of the derivation are provided in the Appendix Section B):
O
At i
At iE
P
V1
V2
V3
At ko [A2 X1)
(1+ At Ko (A3 -11) Al ihy
E EC3x3
Al icy
(1 +1+ A (
(1+ A1)
X1)
Leveraging the scan property and the cigendecomposition of the system matrix, we express the update operator as PAP-1, where P contains the eigenvectors and A is a diagonal matrix of cigenvalues. Notably, the inverse p-1 is computed only per training iteration for the entire sequence, thereby reducing the computational overhead in iterative evaluations.
3.3 Stability Analysis
The system exhibits recurrent dynamic structure similar to that in linear state-space models. Specifically, given the eigendecomposition of the state transition matrix A =PPP- the recurrence takes the form
X, Ax Bu, = PAPP Xx,1 Bun
(17)
where PAP-1 denotes the eigendecomposition. Unrolling this recursion yiclds the explicit solution
n-1
X
A BuB
Σ PA*P Bun-k- k=0
(18)
This standard eigenspace expansion of linear recurrences has also been used oscillatory state-space models, such as LinOSS [Rusch and Rus 2025], which focus on second-order ODE dynamics. Here, we adopt similar formulation but extend it to PDE-driven spatio-temporal dynamics in BioOSS.
To ensure stability, all eigenvalues must satisfy |X| < 1. We begin by observing that the real eigenvalue X I+ < 1 is strictly stable for any positive damping coefficient ky 0.
For the complex conjugate eigenvalue pair 22,3, we compute their squared magnitude:
|A2.al 2 = =(b 1 AEh + 1+ Athe,
EE + 9 (1 + AIR (1 Atke.)
(19)
When both damping coefficients positive (A7,, 0 and ko, 0), the first term is larger than 1, while the second term captures the oscillatory contribution induced by the propagation parameter and the spatial frequency content. In the absence of damping, this oscillatory term drive |22,sl above unity at high spatial frequencies, so stability is not unconditional.
Accordingly, we enforce conservative CFL-like sufficient constraint during initialization and training by restricting the effective propagation rate G. relative to At, Ax, and the damping factors. Concretely, we use
Ar V (1+ Atk (1 Atk
(1A Atk?,)
Gj
At
12
(20)
where Ar denotes the fixed spatial step. This bound should be interpreted as an implementation-level stability safeguard (a sufficient condition used via clipping), rather than a proof of unconditional stability over all spatial frequencies Ey).

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3.4 Eigenfrequency Structure and Emergeet Selectivity
The eigenvalue structure of the BioOSS system provides insight into its emergent oscillatory behavior across the spatial grid. Each pair of complex conjugate eigenvalues corresponds to a spatio-temporal mode governed by the local wave parameters and spatial frequency components. While these modes are not tied to individual neurons, their frequency profiles shape the collective dynamics of the p-field. Although eigenvalues are not explicitly computed or used during training or inference, analyzing them post hoc reveals the types of oscillations the model can support after learning.
In discrete-time systems, complex eigenvalues À a ib are typically expressed in polar form as re where r = va² + 82 denotes the magnitude and 0 lan Lan-a(b/a) is the phase angle. This angle is directly related to the angular frequency of oscillation. As complex eigenvalues arise in conjugate pairs, a half revolution (π)) around the unit circle corresponds to the system's sampling frequency Ar. allowing us map the eigenvalue phase temporal frequency as f rAr
For the dominant eigenvalues λ2,3 derived from Eq. (16), this yields the following closed-form expression for the local oscillation frequency at each grid point (i,j):
20,, 53 + E
J.3 lan
(21)
v(1+ Ath? + Atkg,)
1+AIN + 1tBsk?,
This frequency mapping links each spatio-temporal mode's ' oscillatory profile to the local parameters kP, k2 and the wave vector (Ez, Eg). To empirically validate this interpretation, we assess whether neurons initialized with distinet eigenmodes exhibit the expected frequency selectivity.
(a)
(b)
(c)
(d)
Figure 2: (a) Dominant- Dominant-frqquency map: (b) Spectra at region centers; (c) Time traces (last 200 steps); (d) Final p-field. Quadrants are initialized for 0-10, 10-20, 20-30, and 30-40 Hz and each resonates in its assigned band despite identical white-noise input, supporting the eigenvalue-fiequency mapping. Full spectra and time traces are provided in Appendix
We partition 64 x 64 grid into four quadrants and initialize each with eigenmodes targeting different frequency band (0-10, 10-20, 20-30, 30-40 Hz), using Eq. (21). While Eq. (21) defines mapping from eigenvalue phase to frequency using At, this interpretation is only meaningful when At corresponds to a physical time unit. In our synthetic experiments (e.g., white-noise input with frequency-limine limited bands), we set At 10 ms to reflect biologically plausible temporal resolution, allowing the resulting frequencies to be interpreted in Hz. In contrast, for downstream tasks such as those in Section 4.2. Al is treated as unitless index (e.g., one step one hour in the Traffic dataset), and frequency should be understood as relative, dimensionless measure rather than a physical quantity. All regions receive the same white-noise input. Despite identical stimulation, each quadrant exhibits a dominant-frequenne hypothesis that the eigenvalue phase determines local spectral responses, and demonstrates BioOSS's capacity for encoding frequency preferences through spatially structured initialization.
profile matching its assigned band (Fig. Z}. This supports the
These results offer a mechanistic account of how strating that distinct frequency profiles can emerge even without anatomically imposed groupings. The resulting ceqquennc-seelecti regions resemble functional "islands" with coherent internal dy- namics and spectral specialization. While BioOSS does not currently enforce explicit connectivity boundaries, similar modular topologies defined by dense intra-cluster and sparse inter-cluster con- nections-hav been shown to support susf-nnsta wiring-efficient dynamics in both biological and artificial systems Meunier et al. 22010) Liang et al. 22022]
selectivity arises in BioOSS, demon-
7

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3.5 Computational Complexity
The computational complexity of BioOSS scales linearly with the sequence length T, spatial size HW, and number of layers L. Specifically, the inference cost is HWL and the training cost OLLTHW + d)), where d = HW denotes the dimension of the hidden state. detailed derivation of these results is provided in Appendix
4 Results
4.1
Experimental Setting
This paper primarily focuses on the exploration of oscillatory-bused spatio-temporal model and sequential tasks. Therefore, we do not include Transformer models, as they are end-to-end frameworks. BioOSS is evaluated against several representative sequential models with recurrent mechanisms, including the Linear Recurrent Unit (LRU) [Orvieto al. 2023 S5 (Smith al.) 2022]. and LinOSS [Rusch and Rus 2025]. While alternative sequential modeling approaches, such as the Neural Refined Differential Equation (NRDE) [Irie al. 2022] and the Neural Controlled Differential Equation (NCDE) (Kidger et al. 2020). are available, these require solving differential equations at each time step, leading significant computational and time overhead. As such, exclude these models from our comparison to ensure a fair and efficient benchmarking protocol.
Baseline
Nonlinear Components The core BioOSS system linear, but practical implementations include nonlinear components such as multilayer readouts and self-gating blocks in the model head [Rusch] and Rus 2025 These additions are independent of the oscillatory dynamics and are described in Appendix D
Hyperparameters and Environment All BioOSS models employed in the experiments adopt 2D architecture, with the model size varying across tasks. For each task, the learning rate was selected through a grid search to ensure optimal performance. A comprehensive summary of all hyperparameter configurations is provided in the Appendix Section [ D Table All experiments were conducted using JAX for classification tasks and PyTorch for prediction tasks to ensure consistency with established baselines, datasets, and evaluation protocols. Models were trained using the Adam optimizer and executed on a single NVIDIA GeForce RTX 4090 GPU (24 GB memory). Details on model parameters, GPU usage, and memory consumption are provided in Appendix [ Tables and 6
4.2 Evaluation on Time Series Tasks
The benchmark datasets considered in this section use time steps that do not always correspond to fixed physical durations. For example, in the Traffic dataset, each step represents one hour. As a result, while BioOSS captures oscillatory dynamics across time steps, no mapping to physical frequency units (e.g., Hz) is applied in these experiments. We evaluate the proposed BioOSS model on two fundamental time series tasks: classification and prediction. Experiments are conducted on recent benchmark datasets demonstrate the effectiveness of the bio-inspired architecture.
All models were trained under identical settings, with results averaged over five seeds and 70:15:15 train/validation/test split. For BioOSS, we tuned the learning rate, while for LinOSS and other baselines we relied on the official implementations [Rusch and Rus 2025] without grid search. We note that our reproduced LinOSS results differ slightly from those reported in the LinOSS paper, which we attribute to environment dependencies, lack of hyperparameter retuning, and the strong sensitivity of UEA and PPG datasets to the exact random split across Python/JAX environments. Our aim was not to re-evaluate LinOSS, but to benchmark all models under a unified protocol, presenting BioOSS as a competitive and interpretable alternative rather than new state of the art.
4.2.1 Classification
We evaluate the proposed BioOSS model on recently introduced sequential benchmark presented in Walker al. 2024), using the same benchmark protocol as in the LinOSS paper [Rusch and
Rus 2025 ensure fair comparison and evaluation. This benchmark includes six multivariate time

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series datasets from the University of East Anglia (UEA) Multivariate Time Series Classification Archive (UEA-MTSCA) [Bagnall al., 2018:: EigenWorms (Worms), SeliRegulationSCP! (SCP1), SelfRegulationSCP2 (SCP2), #ttaaollooncentratio (Ethanol), Heartheat, and Motorlmagery (Motor). These datasets exhibit a wide range of sequence lengths and classification complexities, providing comprehensive evaluation of model performance under varying temporal and structural characteristics.
The results are summarized in Table
Worms
SCP1
SCP2
Ethanol
Heartheat
Motor
Seq. length
17.984
896
1,152
1,751
405
3,000
#Classes
5
2
2
4
2
2
LRU
89.4 + 5,4
86.8 + 3.1
54.7
+ 10.1
28.4 + 5.0
70.0
= 3.6
55.8 + 6.2
S5
83.9 + 4.1
86.8
+ 4.5
51.9
+ 4.5
24.3
+ 4.8
73.9 + 3.1
50.5 + 4.2
LinOSS
92.2 + 7.7
85.9
2.9
60.4
+ 5.7
29.9
+ 0.6
71.9
2.6
60.4 + 7.2
BioOSS
92.8
5.2
85.6
+ 3.9
55.1
+ 1.8
33.4
10.7
74.8
2.0
55.8 + 5.8
Table I: Multivariate long-term time series classification results. Accuracy (%) and standard deviation for each model on six UEA datasets. Best results are shown in red and bold; second-best results are in blue and underlined. Higher is better.
We report the multivariate time series classification performance of all models across six benchmark UEA datasets in Table On average, BioOSS achieves the highest accuracy at 66.25%, outperforming all baselines, followed by LinOSS 65. 12%. BioOSS achieves state-of-the-art results on three datasets ( (Worms, SthanolConcentrationn, and Heartbeat) and secures the sccond-best performance on two others (SCP2 and Motorlmagery).
Notably, on the challenging Hcartbeat dataset, BioOSS attains the best accuracy of 74.8%, improving upon LinOSS and S5 by 2.9% and 0.9%, respectively. On the long-sequence EthanolConcentratiion dataset (1,751 steps), BioOSS also outperforms all baselines, achieving 33.4% accuracy compared to 29.9% for LinOSS and 28.4% for LRU.
In contrast, S5 shows competitive performance only on SCPI and Heartbeat, while LRU performs best only on SCP1 and exhibits high variance across datasets. LinOSS demonstrates consistent robustness, achieving the highest accuracy on SCP2 (60.4%) and the second-best result on Worms (92.2%). These results highlight the strong generalization ability of BioOSS, particularly for long sequences and heterogeneous biological and environmental signals.
4.2.2 Prediction
We evaluate the proposed models on the PPG-Dal iA dataset, a benchmark for heart rate regression using wearable sensor data. The dataset contains multi-channel physiological and motion signals collected from fifteen subjects engaged in daily activities. Each subject's 's recordings span approxi- mately 2.5 hours at 128 Hz, covering six input channels: blood volume pulse, electrodermal activity, body temperature, and triaxial acceleration. To handle the long-range dependencies, data using sliding window of length 49,920 and step size 4,992. Following the same evalua- tion protocol in the LinOSS paper [Rusch and Rus 2025 all models are trained with identical hyperparameter tuning procedures to ensure fair comparison and evaluation. Table 2 summarizes
segment the
Model
MSE x10
LRU
15.64 + 1.67
S5
12.79 0.72
LinOSS
6.4 + 0.23
BioOSS
7.7 - 0.2
Table 2: PPG-based regression results on the PPG-DaLiA dataset. Mean-squared error (x (x10-2) and standard deviation over 5 training runs for each model. All models are trained using the same hyper-parameter tuning protocol to ensure fair comparison. Best results are shown in red and bold and second-best results are in blue and underlined. Lower better.
the average test mean-squared enror (MSE) over five independent runs. LinOSS achieves the lowest MSE, while BioOSS secures the second-best performance. Compared traditional recurrent and state-space baselines such as LRU and S5, both LinOSS and BioOSS demonstrate clear advantages

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in modeling long-dunn dungtourtioon dependencies, highlighting the effectiveness of the proposed structured designs for physiological signals.
To further evaluate predictive capabilities, we assess the models on four additional benchmark datasets characterized by clear periodic patterns: Electricity, Solar-Energy. Traffic, and Weather (Lai et al. 2018). Dataset splits, look-back sequence lengths, and prediction horizons are aligned with the cyclie properties of each dataset. The results, summarized in Table further validate the strong performance of LinOSS and BioOSS across diverse long-term forecasting tasks.
Electricity
Solar-Energy
Traffic
Weather
Timesteps
26,304
52,560
17,544
52,696 21 10 mins Daily
Channels
321
137
862
Frequency
hour Daily & Weekly
10 mins
hour
Cyclic Patterns
Daily
Daily & Weekly
Cycle Length
168
144
168
144
LinOSS 21.875 + 0.000
22.363 + 0.000
68.828 + 0.002
22.288 + 0.000
BioOSS
18.699 0.000
21.873 + 0.003
62.284 + 0.007
22.232 + 0.000
Table 3: Multivariate time-series forecasting results across four benchmark datasets with intrinsic patterns. Mean-squared enror (MSE x x10-3) and standard deviation over 5 training runs are reported for each model. All models are trained using the same hyper-paramest tuning protocol to ensure fair comparison. Best results for each dataset are highlighted in red and bold. Lower is better.
Table 3 reports the average test MSE over five runs for LinOSS and BioOSS on four multivariate time series benchmarks. BioOSS achieves lower MSE than LinOSS on all datasets, indicating more effective modeling of temporal dependencies and periodic structure with comparable model size.
4.3 Oscillatory Dynamics Visualization
To qualitatively examine the internal dynamics of BioOSS beyond benchmark accuracy, we generated visualizations of emergent oscillatory behavior in the 2D oscillatory layer. Specifically, normalized logits were extracted from an audio tagging backbone processing a 1-minute audio segment using 4-second sliding window with I-second stride. These logits were then used as point-wise inputs to BioOSS, with each logit dimension mapped to local patch of 7-8 grid cells. The system subsequently evolved under BioOSS dynamics, and the pressure-field (p) values were recorded over time. The resulting sequences reveal coordinated wave-like spatio-temporal motifs: activity initiated in one region induces structured oscillatory responses in neighboring regions under the learned grid dynamics. This dynamic behavior complements the static examples shown in Fig. dd-e), providing direct evidence of emergent spatiotemporal coordination. Animated GIFs of representative sequences provided in the Supplementary Material, which contains three audio samples illustrating the evolution of the p-field under realistic inputs.
5
Conclusion
We proposed BioOSS, a biologically inspired oscillatory state system that captures spatio-temporal dynamics through wave-inspired, oscillation-driven coordination between pressure-like and velocity- like states on a 2D grid. By introducing trainable damping and propagation parameters, BioOSS adapts to task-specific patterns while preserving numerical stability and interpretability. It achieves strong performance across classification and forecasting benchmarks, particularly excelling in frequency- selective and long-range temporal tasks. Unlike traditional models, BioOSS offers insight into its internal dynamies via eigenvalue-baser frequency decomposition. While promising, future work can explore scaling, multimodal integration, and continuous learning. BioOSS offers a step toward interpretable, brain-inspired computation in AI. detailed discussion of limitations is included in Appendix F
10

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Acknowledgment
This work was supported by the Special Research Fund (BOF) of Ghent University under Grant BOF/24//2021/246 and by the Flemish Government through the Onderzoeksprogrammaa Al Vlaan- deren programme. We gratefully acknowledge the valuable effort of Prof. Eamonn Keogh and his team in creating, curating, and sharing the datasets used in this study.
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12

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A Fourier Transformation of Differential Operators
In this section, derive the fundamental relationship between differential operators in physical space and their counterparts in Fourier space.
A.1 Definitions and Notation
We begin with the definition of the Fourier transform for function p(r) in RR:
( F(p(x)) = = p(r)e- if dr,
(22)
where ş = [Ezz .. the frequency domain vector.
For simplicity, we focus on the two-dimensional case where (ar, v) and & [Sxx Eyl
A.2 Transformation of the Gradient Operator
We aim to derive the Fourier transform of the gradient of p. denoted by Vp:
F(Vp) = F
(23)
We first evaluate the Fourier transform of the spatial derivative of p. given by F { dv
{ Op
Öp(z. v) e ((((x, da dy.
F
(24)
0x
dr
gaa
Focusing on the integration with respect to we apply integration by parts: (e (e=t)x)) dx
|0 Öp(z,3) 3)
i8.= dx = [p(x, y)e
10. --x
p(z. v) dx
dx
= [p((,, y)
1200000 p(z, v) -if de (25)
=pp(z,y) (600 + its I p(a, y)e 16.0
-ix dr.
For functions that are sufficiently smooth and decay rapidly as |x| - oc, the boundary term vanishes: [p(x, y)e [Ea] x
0.
(26)
Therefore:
th Op(z,
- dx = it / p(x, y)e 1hx
-ix dx.
(27)
dr
Reintroducing the integration with respect to y:
Op
0p(z, -((fxx+byV) dr dy
F
0x
dr
iE.
p(z, y) 15.0 dx -ssu dy
(28)
iL.
p(x, w)e -llxxtbuv)) dx dy
-00
iE.p(Exx Eg).
By symmetry, we can similarly derive:
Op
iEg,p(Ee En))
(29)
0y
13

---

Thus, for the gradient operator, we have:
Op -(Vp)=
1 Lyy
iEp.
(30)
By
This gives the important relationship:
Vp - iEp
(31)
A.3
Transformation of the Divergence Operator
For vector field o [0x; 0y]", T, the divergence operator is defined as follows:
0o 8oy
V
(32)
dr
Oy
Applying the Fourier transform and using the result from the previous section:
do
dog
F(V o) =
0x
Oy
{
00
Doy
+
(33)
da
i5
iEngo
= ig ö.
This gives the important relationship:
o E ig ö.
(34)
Thus, the Fourier transform converts differential operators into algebraic operations:
0 J, iw.
(35)
Øt
Vp 7
iEP
(36)
o - it ô.
(37)
B Analytical Form of Eigenvectors P
We consider the linearized local system matrix at a grid point (i.j) in Fourier space:
a
-ac² At
-ac² At iEy
1
A
BAL i..r 8
0
B
A
1 Ao
BAL iEy
3
Let À be an eigenvalue of A, and v = [U1, U2 Us its associated eigenvector. We assume without
loss of generality i1 and solve (A AIvv 0. From the second and third rows, obtain
3 iE.r
31 iE,
A B
A- B
For eigenvalues A2 and A3 corresponding to oscillatory modes, the associated eigenvectors are
At iE,
At iE
V3 = (1 At A0 (Ag -1) V2 + At ko )(2 - A1)
At i
it,
(1+ At k k²)(A2- A1) -1)
(1 At kt) -
In contrast, for the purely dissipative eigenvalue A = 8, the eigenvectors correspond to vectors in the nullspace of the first row. A basis can be chosen, for example, as
0
0
1
0
0

---

C
Computational Complexity Analysis
The computational cost of BioOSS primarily arises from its spatio-temporal update in the hidden layer, which is based on training and inference complexity as follows.
approximations of a coupled PDE system. We characterize
Let E R and o = (o", )²) E [B2 denote the pressure- and velocity-like states defined on 2D grid. At each time step, BioOSS layer performs:
Gradient update:
Pi. - Pi-1.5
Pi.j - Pi.j-1
V Pi.j
A
A
Divergence update:
0f
0 b,++1
o%
1+1
0
+
Az
Az
Damped update:
(
o²
p(
p
1+ At ko
1 At KP
These operations require (HH) computations per layer per time step, where H W the spatial resolution of the 2D state. Each layer also performs linear projections, gating, and readout on hidden state of dimension = HWW yielding a cost of OHW -d).
Thus, the total inference cost per layer over I'' time steps is:
OLTHHWd),
During training, gradients must also be computed through the differentiable spatial operators and projection layers, with weight updates costing O(HW dd²) per layer. Therefore, the total training complexity is:
OLT + +))..
In practice, the hidden layer dominates both inference and training cost, and the model's complexity scales linearly with sequence length T, number of layers L, and spatial size HW.
D Training Details
Algorithm Iillustrates the full architecture of the multi-layer BioOSS model. While the main text focuses single-layer formulation, the full model stacks L. such layers sequentially. Each layer integrates two key components: a biologically inspired spatio-temporal propagation mechanism and scan-compatible gated update.
Given an input sequence u, the hidden state z is first updated by solving a PDE-based wave propagation equation over 2D neural grid, as defined in Eq. B This involves local interactions between pressure-like neurons and velocity-like neurons through spatial gradients and divergence operators. These updates are governed by trainable wave speed e and damping coefficients kP, ko.
The resulting hidden state is then projected through linear layer and passed to a gating mechanism, inspired by LinOSS Rusch and Rus [2025), but applied on 2D spatio-temporal field. The gating update follows:
cu o aW, o tanh(W...0)) + (1 oW, ()) x(-11)
where W3 and W, are learnable matrices shared across time. This residual formulation promotes stable deep learning and supports efficient scan operations.
Next, the gated output is mapped via linear readout: (0) = Ca. + (u(1-1), followed by GELU activation Hendrycks and Gimpel (0066). A Gated Linear Unit (GLU) Dauphin et al. [2017) then modulates the activations via GLU (Wi (War). The result is added to the previous layer input to produce (!),, which is fed into the next block.
(1)
15

---

Algorithm Training Procedure for Full BioOSS Model with Spatio-Tempon Layer
1: Input: Input sequence u
2: Output: BioOSS output sequence after L layers
3: for 1 to L do
4: // Step Spatio-temppral PDE-based propagation over 2D grid
5: Compute gradient: Vp! + SpatialGradien((""))
6:
Compute divergence: V (!-- + SpatialDiveggenc(((o-])
Update velocity-like neurons: o" + o!! - At Vp((-1)
7:
8:
Update pressure-like neurons: p" < ((1-1) At (c2 @ V oll-1) + At. Bu(-1)
9:
Apply damping: 00 + o" 001(1 [1 At k²), pl) t p' /11 kt.
10:
Hidden state: x²()) < concat(pl)) ,(())
11:
// Step 2: Linear projection and gated residual update (SSM-style)
12:
Linear projection: 2(() +- Wax(()
13:
Gating: gate!!) +- o(W
Residual gated update: x gate taan(= + gate ((--1)
14:
15:
// Step 3: Readout and layer update
16:
Linear readout: 7 &- Cz + (u(
17:
Apply nonlinearity: y( t- (E)U(y
18:
Update input for next layer: (!)) + GLU()!)) + u(t
19:
end for
20:
Return final output: o W Wouty(L) + bont
After L layers, the final output computed as o Wonty( + bout- All operations are parallelized across time steps, and time indices are omitted from the notation for clarity.
The best-performing hyperparameter configurations for the BioOSS model across all datasets summarized in Table 4 These configurations were obtained using the same search procedure described in the main paper (Section 4.2p. Due the high nonlinearity of the BioOSS model, its performance is highly sensitive to both the learning rate and the spatial grid resolution. Therefore, for each dataset, we conducted systematic grid search, exploring learning rates from 0.000001 to 0.005, hidden dimensions of 16, 32, 64, 128, and 256, and spatial grid sizes of 10x 10, 15x 15, and x0. For each dataset, we report the selected learning rate, hidden dimension size, spatial grid resolution. and number of blocks. Notably, single block was sufficient for all cases, while the optimal learning rate, hidden size, and grid size varied depending on the dataset characteristics.
Model Worms
Ir
Hidden Dim
Grid Size
#layers
0.0001
128
20 20
SCP1
0.0002
128
20 x 20
SCP2
0.00001
128
15 15
Ethanol
0.00025
16
20 x 20
Heartbeat
0.000005
16
20 20
Motor
0.000004
128
20 x 20
PPG
0.0002
128
20 20
Electricity
0.01
16
10 X 10
Solar-Energy
0.01
16
10 10
Traffic
0.01
16
10 X 10
Weather
0.01
16
10 10
Table 4: Best hyperparameter configurations for BioOSS.
We report the number of parameters, GPU memory usage (in MB), and run time (in seconds) for all evaluated models across the datasets introduced in Section 4.2 The experimental settings for the baseline models strictly follow those established in [Rusch and Rus 2025), ensuring consistency in both codebase and hardware environment. All experiments were conducted on the same GPU device to guarantee fair and direct comparability. Table summarizes the model configurations, memory consumption, and training run time for the time-series classification (TSC) and PPG forecasting tasks, where all experiments were implemented using JAX. In contrast, Table presents the corresponding
16

---

results for the long-term seme-series forecasting (TSP) benchmarks, with experiments conducted in PyTorch. We note that differences in parallel computation mechanisms between JAX and Py PyTorch may account for minor variations in GPU memory usage. Overall, LinOSS and BioOSS exhibit competitive memory efficiency and favorable run time performance relative established baselines such as LRU and S5.
Dataset
Config
LRU
S5
LinOSS
BioOSS
#parameters
101129
22007
134279
153549
Worms
GPU memory (MB)
18761
18877
18747
18727
run time (s)
138
78
11
15
#parameters
25892
226328
991240
153162
SCP1
GPU memory (MB)
18759
18879 4
18893
18739
run time (s)
15
34
6
#parameters
26020
5652
448072
107002
SCP2
GPU memory (MB)
18781
18881
18853
18741
run time (s)
10
46
#parameters
76522
76214
6728
16332
Ethanol
GPU memory (MB)
18737
18915 5
18813
18733
run time (s)
9
#parameters
338820
158310
10936
16300
Heartbeat
GPU memory (MB)
18503
18885
18835
18741 2
time (s)
#parameters
107544
17496
91844
153162
Motor
GPU memory (MB)
18753
18857
18769
18727
run time (s)
26
#parameters
107544
17496
91844
169416
PPG
GPU memory (MB)
18773
18915
18719
18729
run time (s)
107
57
21
17
Table 5: Number of parameters, GPU memory usage (in MB) and run time (in seconds) for selected models on all datasets from Table and Table 2 in Section 4.2
Model
Config
Electricity
Solar-Energy
Traffic
Weather
#parameters
91521
67785
161310
52821
LinOSS
GPU memory (MB)
1871
1717
2079
1725
run time (s)
89
70
105
71
#parameters
101 129
22007
26119
134279
BioOSS
GPU memory (MB)
1765
1659
1967
1653
time (s)
170
170
196
162
Table Number of parameters, GPU memory usage (in MB) and run time (in seconds) for selected models on all datasets from Table 3 in Section 4.2
E
Additional Spatio-Temporaa Dynamics Related Results
In this experiment, spatially partitioned 64 64 BioOSS structure was stimulated with white noise inputs of varying frequency bands (0-10 Hz, 0-20 Hz, and 0-30 Hz) to investigate its frequency- selective response properties, as shown in Fig. 2 Each quadrant of the grid was designed exhibit distinct natural frequency preference. The dominant frequency response maps (Figs. 3a 3e and 31) reveal that each spatial region selectively responds to components of the input signal that match its predefined natural frequency. When the input white noise was limited to the 0-10 Hz band (first row), all regions exhibited oscillations concentrated around 10 Hz, as no higher frequency components were available to activate segiifc resonances. As the input bandwidth increased to 0-20 Hz. (second row), the bottom quadrants began to resonate at their natural frequencies within the lower band, while the top quadrants, unable to find matching excitations at their designated higher natural frequencies,
17

---

adapted by oscillating around the upper limit of the input band near 20 Hz. This behavior is clearly observable the frequency spectra and time-domain responses selected points (Figs, 31 and 3g) Finally, with a 0-30Hz input (third row), the top left quadrant was also able to resonate at its natural frequency within the newly available band (Figs. 3k whereas the top right quadrant continued to oscillate predominantly around 30 Hz, indicating that its natural resonance lies beyond 30 Hz and thus remained driven by the highest available frequencies. The final pressure field snapshots (Figs, [d 3h and 31) illustrate the emergent spatial patterns after 2,000 time steps for each input condition. These observations demonstrate the spatial-frequency selectivity the BioOSS and its capacity to filter and respond dynamically to broadband excitations in region-specifie manner.
(a)
(b)
(c)
(d)
(g)
(k)
Figure 3: Spatially partitioned frequency-sellective responses of the 64 64 BioOSS under varying white noise input bands. The three rows correspond to different input frequency bands: (a)-(d) 0-10 Hz, (e)-(h) 0-20 Hz, and (i)-() 0-30 Hz. Within each row: (a, depict the spatial distribution of dominant frequencies across the four quadrants the grid, illustrating distinct regional frequency responses despite identical white noise excitation; (b, f. j) show the frequency spectra at the center of each region, highlighting region region-specific resonant behavior; (c, g. k) present the time-domain responses at selected points during the final 200 time steps, reflecting the differentiated temporal dynamics induced by the requeecyyselectiig properties of each partition; (d, h, 1) display the final pressure field P after 2,000 time steps, revealing the emergent spatial patterns shaped by the respective input conditions.
F
Limitation
The model's non-linear dynamics introduce mild training complexity and hyperparameter sensitivity. Morcover, the linear encoder limits the expression of intrinsic oscillatory behavior, making the model partial abstraction of biological signals.
18

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NeurIPS Paper Checklist
Claims
Question: Do the main claims made in the abstract and introduction accurately reflect paper's contributions and scope?
Answer: [Yes]
The abstract and introduction clearly state the paper's main contributions, including the development a bio-inspired model to capture the sptio-temporal dependency, its integration with oscillatory dynamics, and its application to range of temporal tasks. These claims are consistently supported by both theoretical insights and experimental results throughout the paper. The scope of the work are also appropriately framed, ensuring that the reader has a realistic understanding of the model's s capabilities and generalization poten- tial. The narrative in the introduction aligns with the broader motivation while accurately delineating the specific contributions made.
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: The paper includes dedicated "Limitations" section that discusses the main assumptions underlying our approach and how they might affect the generalizability of the results. Additionally, we point out potential sensitivity to hyperparameter settings. By providing a separate section for limitations, we aim to ensure transparency and help readers understand the scope and boundaries of our claims.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-ter system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they seale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimess.
19

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While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [Yes]
The paper includes several theoretical results, each accompanied by a clearly stated set of assumptions and formal statements. All theorems, lemmas, and key formulas are properly numbered and cross- cross-referenced throughout the manuscript. Complete and correct proofs are provided the appendix, and for results whose proofs appear in the supplemental material, intuitive proof sketches are included in the main text to aid understanding. Further- more, all supporting theorems and lemmas used in the proofs are appropriately cited and referenced, ensuring clarity and correctness in the theoretical development.
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
The paper provides all necessary details reproduce the main experimental results, including descriptions of the datasets, model architecture, training procedures, and evaluation protocols. All hyperparameters are specified, the experiments use public benchmark datasets.
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Reviews:
[{"review_id": "GS4tlF0iq0", "reviewer": "Reviewer_xBBL", "summary": "This paper builds on current literature on biologically inspired models, specifically on the recent LinOSS paper and proposed a new mathematical formulation for oscillatory neural networks, The paper claims this current formulation is better at spatio temporal correlations and shows results on several benchmarks", "questions": "How do structured local interactions among p and o neurons influence the emergence of wave-like propagation patterns, and how critical are these interactions for encoding temporal dependencies?\n\nHow robust is BioOSS to noisy, sparse, or irregularly sampled time-series data compared to other oscillatory or recurrent models?\n\nCan you characterize mathematically the computation requirements for training and inference?\n\nCan you state clearly where does BioOSS clearly outperform standard models—and why?\n\nIs the interpretability gained from spatial frequency structure usable in downstream applications like clinical diagnostics or robotics?", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros\nThe model and the discretization formuation are nice. \nThe explicit modeling of both spatial and temporal interactions allows for richer dynamics than standard sequence models, capturing structure closer to real neural circuits.\nBioOSS improves transparency and analysis of internal representations\nThe paper moves forward the field of oscillatory models.\nThe paper presents a reasonable suite of benchmarking tests.\n\nCons\nWhile biologically inspired, the model’s structural complexity (2D grids, dynamic interactions) may pose challenges for scaling to large data or high-resolution domains.\nComparisons to state-of-the-art deep models (transformers & SSMs) in high-impact benchmarks are needed to demonstrate practical utility. It would be great to see comparisons to Mamba\nIt would be great to go deeper into the mathematics of these models and identify beyond empirical results on benchmarks (because a benchmark is just a benchmark and BioOSS does not always have the best performance on the benchmarks) why and when this mathematical model will provide effective spatio-temporal correlations\nMore empirical work across diverse domains (e.g., vision, speech, control) would help ground claims of generalizability.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for the insightful feedback, which helped us clarify key aspects and identify promising future directions.\n\n**Q1. Comparisons to state-of-the-art models (e.g., Transformers, Mamba) are needed to demonstrate practical utility and clarify when BioOSS effectively captures spatio-temporal correlations beyond benchmarks.**\n\n**A1.**\nWe used the same setup as LinOSS for fair comparison among SSMs. In this configuration, Mamba underperformed S5 and LinOSS, so its results were omitted. On periodic time series datasets (Table 3), BioOSS does not outperform SOTA Transformers, though results remain acceptable. We thus focus comparisons within the SSM context. Moreover, MSE may not fully reflect model performance. For example, on regular rhythm datasets, BioOSS shows more biologically plausible dynamics (see analysis in A3).\n\nRather than chasing benchmarks, our goal is to propose a biologically grounded, interpretable framework based on 2D oscillatory wave propagation. Unlike Mamba and Transformer models, whose core building blocks are perceptron-style transformations with pointwise activations, BioOSS is governed by underdamped PDEs simulating traveling waves and resonant spatiotemporal behavior.\n\nFrom a neuroscience perspective, mixed selectivity is a core principle of computation in cortical circuits [1], where diverse oscillatory responses arise from complex neural connectivity and are combined downstream for decision making and motor control. In this context, BioOSS’s oscillatory dynamics offer a biologically faithful substrate for learning. Resonant modes encode a distributed basis of temporal and spatial features, whose superposition enables flexible, compositional input representations.\n\nThis form of dynamic mixed selectivity is difficult to achieve with purely feedforward, perceptron-based models like Mamba or Transformers. Therefore, BioOSS offers a novel approach not just in terms of empirical benchmarks, but in its capacity to model the rich dynamical substrate of biological computation.\n\n**Q2. How do structured local interactions among p and o neurons influence the emergence of wave-like propagation patterns, and how critical are these interactions for encoding temporal dependencies?**\n\n**A2.** \nSpecifically, the gradient of $p$ drives local changes in the velocity field $o$, while the divergence of $o$ determines how the pressure field $p$ evolves across the grid. These spatial operators are applied across a 2D lattice, allowing perturbations at a single point to propagate to neighboring nodes, thus forming traveling wavefronts.\n\nThe local interactions between $p$ and $o$ neurons are essential: the gradient of $p$ drives flow, and the divergence of $o$ redistributes activity across the grid. Together, these operators form a locally coupled system that supports oscillations, rather than pointwise recurrence.\n\nCrucially, these oscillations persist in the system even after external input vanishes, due to the underdamped nature of the coupled PDE. This enables short-term memory to be maintained intrinsically by the system’s internal dynamics, without requiring explicit state storage or gating. Instead of vanishing instantly, the residual oscillations encode a compressed trace of past inputs, which can be retrieved and transformed by downstream layers.\n\nIn this way, the $p$–$o$ coupling is not only responsible for spatial signal propagation, but also acts as a mechanism for distributed temporal memory, where oscillations serve as transient carriers of recent history. To preserve the physical consistency of the FDTD formulation and maintain SCAN-compatible discretization, we did not ablate the $o$ field under the SSM framework. Instead, we introduced the hidden layer into an RNN setting and conducted a single-field ablation on visual and auditory tasks (see response to p2sQ for details).\n\n**Q3. How robust is BioOSS to noisy, sparse, or irregularly sampled time-series data compared to other oscillatory or recurrent models?**\n\n**A3.** \nBioOSS is expected to exhibit robustness to sparse or irregularly sampled inputs due to its oscillatory dynamics and spatially coupled structure. In particular, the damping term $k$ stabilizes the system, while the spatial gradient and divergence operators ensure distributed propagation of information across the grid.\n\nAlthough this work does not explicitly evaluate irregular sampling, we previously applied a similar hidden layer to beat-based rhythm prediction tasks where the input time series was sparse, non-zero only at beat onsets, and zero elsewhere. Only our model retained beat memory after input ceased, producing an extra peak at the expected beat time. In contrast, baseline models such as LSTMs and Transformers failed to extrapolate beyond observed beats and did not show such predictive peaks.\n\nThese findings support the hypothesis that BioOSS’s oscillatory memory and spatial redundancy confer advantages in handling sparse temporal data. We see this as a promising direction for future work.\n\n**Q4. Can you characterize mathematically the computation requirements?**\n\n**A4.**\nYes. The computational cost of BioOSS primarily arises from its spatio-temporal update in the hidden layer, which is based on finite-difference approximations of a coupled PDE system. We characterize training and inference complexity as follows.\n\nLet $p \\in \\mathbb{R}^{H \\times W}$ and $o = (o^x, o^y) \\in \\mathbb{R}^{2 \\times H \\times W}$ denote the pressure- and velocity-like states defined on a 2D grid. At each time step, a BioOSS layer performs:\n\n(a) Gradient update\n$\\nabla p_{i,j} = \\left( \\frac{p_{i,j} - p_{i,j-1}}{\\Delta x},\\ \\frac{p_{i,j} - p_{i-1,j}}{\\Delta x} \\right)$\n\n(b) Divergence update\n$\\nabla \\cdot o_{i,j} = \\frac{o^x_{i,j+1} - o^x_{i,j}}{\\Delta x} + \\frac{o^y_{i+1,j} - o^y_{i,j}}{\\Delta x}$\n\n(c) Damped update\n$o^{(l)} = \\frac{o^\\ast}{1 + \\Delta t \\cdot k_o},\\quad p^{(l)} = \\frac{p^\\ast}{1 + \\Delta t \\cdot k_p}$\n\nThese operations require $\\mathcal{O}(HW)$ operations per layer per time step, where $H \\times W$ is the spatial resolution of the 2D state.\n\nEach layer also performs linear projections, gating, and readout on a hidden state of dimension $d=HW$ (number of $p$ neurons), yielding a cost of $\\mathcal{O}(HW \\cdot d)$.\n\nThus, the total inference cost per layer over $T$ time steps is:\n$$\n\\mathcal{O}(L T H W d)\n$$\n\nDuring training, gradients must also be computed through the differentiable spatial operators and projection layers, with weight updates costing $\\mathcal{O}(HW \\cdot d^2)$ per layer. Therefore, the total training complexity is:\n$$\n\\mathcal{O}(L T H W (d + d^2))\n$$\n\nIn practice, the hidden layer dominates both inference and training cost, and the model’s complexity scales linearly with sequence length $T$, number of layers $L$, and spatial size $HW$.\n\n**Q5. Can you state clearly where BioOSS clearly outperforms standard models—and why?**\n\n**A5.**\nOur primary objective is not to make incremental gains on benchmark performance, but to propose a biologically grounded, physically interpretable, and generalizable modeling framework based on oscillatory dynamics. BioOSS is designed to address three core challenges: interpretability, trainability of physically inspired systems, and architectural generalizability.\n\n1. Interpretability via Biophysical Modeling\n\nBioOSS simulates spatio-temporal neural activity on a 2D grid, analogous to the layout of cortical tissue. The dynamics are governed by finite-difference approximations of wave propagation (FDTD2D), reflecting how neural oscillations travel across brain tissue. This structure enables spatial frequency analysis and transparent interpretation, unlike abstract Transformer or RNN representations.\n\n2. Trainability of Physically Realistic Oscillatory Systems\n\nUnlike traditional models that are mostly based on perceptual or spike-based neuron abstractions, BioOSS employs a coupled PDE system with damped wave dynamics. While previous physically grounded models are difficult to train due to instability in backpropagation (especially through implicit solvers), our explicit Euler scheme ensures stability and differentiability for gradient-based training.\n\n3. Generalizability beyond PDE Simulation\n\nFDTD systems are typically used in constrained simulation settings and lack architectural flexibility. BioOSS overcomes this by embedding the wave dynamics as a differentiable hidden layer, enabling integration with SSM, RNN, or even attention-based architectures. Here, we focus on its SSM role for long-range modeling, where it matches SOTA models like S5 and LinOSS and outperforms them on harder datasets. However, the structure also supports variants with CNN or attention-based couplings, which we will test in separate experiments.\n\nIn sum, BioOSS does not aim to outperform every model on benchmark scores universally. Rather, it contributes a new modeling paradigm that is biophysically interpretable, mathematically grounded, and practically trainable, while maintaining competitive performance on general long-range sequence tasks.\n\n**Q6. Is the interpretability gained from spatial frequency structure usable in downstream applications like clinical diagnostics or robotics?**\n\n**A6.**\nYes, BioOSS’s frequency selectivity enables interpretability. We extract neuron-wise frequency responses from the eigenvalue spectra, revealing dominant low-frequency modes consistent with oscillatory behavior. These signatures could be used for downstream tasks such as identifying frequency-based biomarkers in neuroimaging or decoding sensory information in robotics.\n\nThough our paper focuses on benchmarks, these dynamics have potential in broad downstream tasks. Future work will explore these directions.\n\n[1] Rigotti, M., Barak, O., Warden, M. R., Wang, X. J., Daw, N. D., Miller, E. K., & Fusi, S. (2013). The importance of mixed selectivity in complex cognitive tasks. Nature, 497(7451), 585–590."}, {"review_id": "YZ0I0o0FvF", "reviewer": "Reviewer_NXt7", "summary": "The paper introduces BIoOSS, an extension of the recently proposed LinOSS model that incorporates locally coupled dynamics into the original framework.The paper provides an explanation how parallel scans can still be used for this framework and finally presents numerical results of BIoOSS mostly comparing it to the LinOSS model. However, the manuscript suffers from poor organization, lacks a clear motivation for introducing local coupling, and fails to demonstrate improved performance—especially when benchmarked against the originally published LinOSS results rather than the modified baselines reported in this paper (more in strengths and weaknesses).", "questions": "See weaknesses", "limitations": "The model might not scale well with large number of nodes. It does not provide any clear advantages over the original LinOSS model and performs similarly or worse in practice.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n* the paper extends a recently published model that has gained significant attention \n* the paper provides empirical results on various datasets\n\nWeaknesses:\n* The paper clearly extends and is fully based on the recently published LinOSS model of Rusch and Rus, 2024, e.g., the whole structure and representation follows the LinOSS paper almost verbatim. However, this is not explicitly stated and LinOSS is only cited twice before section 4 in the context of \"other oscillatory models\". I urge the authors to provide proper credit and clearly state that this works follows closely Rusch and Rus, 2024. \n\n* While BIoOSS does introduce coupling between nodes, the explicit formulation used in this manuscript lacks proper motivation. Why is this type of coupling used and not anything else? This is never explained nor motivated.\n\n* It's not clear why coupling is needed for the LinOSS model. Rusch and Rus, 2024 have shown that LinOSS is universal. What functions/operators can BIoOSS express that LinOSS cannot? I presume there is none. Moreover, LinOSS does have coupling mechanism (otherwise it would not be universal). However, this is done in between layers, i.e., the dense input weight matrix modulates and couples the outputs from the layer before, which by definition is a coupling mechanism. Why is that not sufficient? There needs to be more explanations and analysis.\n\n* The writing of the paper is poor, e.g., it's not clear how the underlying system is solved using parallel scans. In line 131-134: \"This associativity is essential because it enables the use of parallel scan operations, reducing the computational complexity [...]. Therefore, we need to make the eigendecomposition [...]\". How does the eigendecomposition help here? I presume the system is solved by diagonalizing it. However, this is never mentioned explicitly. \n\n* Line 155-156 says that P inverse only needs to be computed once. However, doesn't it need to be computed for every node i,j? How costly is that? Do you include it in the speed comparisons in the appendix?\n\n* Table 1 provides weaker results for LinOSS than stated in the official LinOSS paper. This is very problematic. If one considers the originally published results of LInOSS, it performs much better (and not worse as stated in this manuscript) than BIoOSS (67.8\\% for LinOSS vs 66.25\\% for BIoOSS).\n\n* The eigendecomposition provided is only an approximation. There is no further explanation or analysis provided how this affects the model behavior and performance.\n\n* The paper does not provide any deeper or theoretical insights of this model nor lacks a clear motivation of why this particular model is chosen.", "was_revised": true, "final_justification": "I am happy with the changes promised by the authors. This will make the paper significantly stronger.", "rebuttal": "We would like to clarify that BioOSS does not aim to introduce a new oscillation-based state space model (SSM), nor is it framed as a contribution to the SSM family. Instead, we adopt SSM-compatible components as a practical means to explore how biologically grounded dynamics, particularly PDE-driven wave propagation, can be integrated into general-purpose modeling pipelines.\n\nBioOSS is conceptually informed by prior work, especially LinOSS (Rusch and Rus, 2024), as reflected in the scan-based recurrence architecture. We have mentioned LinOSS over twenty times throughout the manuscript, including in the Introduction, Related Work, Methods, and Results. In the Related Work section, we only mentioned the closest model, LinOSS. Some reviewers have also recognized this connection. Although we did not apply hyperlink formatting to every mention, we are happy to provide clearer cross-referencing in relevant sections, such as the scan formulation and algorithm caption.\n\nOur goal is not to reimplement or improve upon LinOSS, but to present a distinct approach grounded in a first-order PDE system that models wave dynamics over a 2D neural grid. This leads to a different form of spatio-temporal recurrence and biologically interpretable behavior, distinct from second-order ODE-based models.\n\nEquations such as (17), (18), and the identity $\\mathbf{P}\\mathbf{P}^{-1} = \\mathbf{I}$ are standard linear algebraic tools included for mathematical completeness and clarity, not specific borrowing. To prevent misunderstanding, we are open to clarifying the presentation or adding attribution notes.\n\nIn the final version, we will further emphasize how BioOSS diverges from LinOSS in its theoretical formulation, spatial structure, and modeling goals, while also improving clarity around citation and attribution.\n\n## 1. Clarifying the Relationship to LinOSS\n\nThe apparent similarity arises from a scan-based recurrent architecture with gating mechanisms. However, we highlight several key distinctions:\n\n- **Shared components are not unique to LinOSS:** Scan operations and gating are widely used in the state-space modeling literature (e.g., Mamba). These components are standard in the field and are adopted in BioOSS purely to ensure a fair and consistent comparison across models, particularly with LinOSS. We would like to emphasize that BioOSS does not claim to propose a novel oscillation-based SSM, nor is it positioned as a contribution to the SSM model family. Instead, our core innovation lies in formulating oscillatory dynamics from a biologically grounded partial differential equation (PDE) framework, which gives rise to distinct spatio-temporal behavior. To facilitate comparison with existing models, including those formulated in SSM form, we express parts of our architecture using SSM-compatible components. This design choice is practical and does not reflect the conceptual foundation of BioOSS, which remains independent of the standard SSM lineage.\n\n- **Fundamental theoretical difference:** LinOSS is based on *forced linear second-order ODEs*, while BioOSS is derived from a *first-order PDE system* inspired by 2D wave propagation (FDTD2D). While LinOSS is a universal approximator of continuous causal operators, universality alone is not sufficient for modeling biologically realistic neural dynamics. BioOSS prioritizes spatially structured, frequency-selective, and interpretable dynamics grounded in cortical circuit motifs, such as lateral wave propagation and localized damping.\n\n- **Spatio-temporal representation:** LinOSS uses 1D hidden vectors with no explicit spatial topology. BioOSS instead models a 2D grid of neurons with local interactions, enabling structured spatio-temporal processing aligned with biological cortical layers and spatial wavefront dynamics.\n\n- **Discretization method differs:** BioOSS discretizes the first-order PDE using a hybrid Euler scheme, commonly used in wave propagation solvers such as FDTD. LinOSS derives its recurrence from implicit and IMEX solutions to a second-order ODE. This leads to distinct numerical properties, inductive biases, and biological interpretability.\n\n- **Different scan application:** LinOSS uses scan along time with global hidden-to-hidden matrices. BioOSS applies the scan over a spatially-coupled operator derived from PDE eigenvalues, governing localized interactions across the grid.\n\n- **Key distinctions in hidden-to-hidden coupling:** In **LinOSS**, the hidden-to-hidden matrix is a dense, learned matrix over feature dimensions, acting globally and uniformly across time. In **BioOSS**, the coupling matrix A governs local interactions across a 2D grid of neurons, derived from a physically grounded PDE and encoding wave propagation properties through meaningful parameters (c and k). BioOSS's recurrence is spatially structured and biologically interpretable, enabling localized oscillations and wavefronts.\n\n## 2. Motivation and Formulation of the Coupling Mechanism\n\n- **Biological inspiration:** Our model introduces distinct *primary neurons* and *intermediate neurons* inspired by cortical architectures. The coupling matrix is derived from FDTD-based wave propagation principles, using physically meaningful parameters: propagation speed (c) and damping (k).\n\n- **Tunable local resonance:** The PDE-derived coupling allows BioOSS to adapt local oscillatory behavior via eigenvalues analytically determined by c and k, akin to tuning a frequency response.\n\n- **Enhanced expressivity:** While LinOSS relies on temporal coupling alone, BioOSS offers spatio-temporal interactions essential for modeling wave-like behavior and frequency-specific propagation relevant in auditory and visual processing tasks.\n\n## 3. Eigendecomposition, Diagonalization, and Scan Efficiency\n\n- **Analytical diagonalization:** Our PDE formulation leads to a structured system matrix that is analytically diagonalizable. Closed-form eigenvalues are derived from c and k via spatial Fourier analysis, avoiding expensive numeric eigendecomposition.\n\n- **Scan efficiency:** To efficiently implement parallel scan operations, we diagonalize the coupling matrix A via eigendecomposition: $ A^n = P \\Lambda^n P^{-1} $. Here, $A$ denotes the trainable spatial coupling matrix governing recurrent dynamics across the 2D grid. The matrix $P$ transforms from the frequency domain to the physical (spectral) domain and is computed analytically, globally defined over the entire 2D hidden layer.\n\n- **Approximation justified:** The eigendecomposition is based on a physically motivated linear approximation of the nonlinear PDE. While approximate, the parameters c and k are trainable, allowing backpropagation to correct for errors.\n\n## 4. LinOSS Performance in Table 1\n\nWhile we made every effort to reproduce LinOSS results using the official codebase and provided hyperparameters, we encountered challenges achieving the same performance. The repository lacks pre-trained checkpoints or explicit data splits, and available logs did not allow full reconstruction of the training pipeline. To ensure fair comparison, we re-evaluated all baselines under a unified pipeline and report the best reproducible results.\n\n## 5. Theoretical Insights and Model Justification\n\nBioOSS is grounded in a biologically interpretable PDE formulation, inspired by wave propagation in continuous media and the spatio-temporal dynamics of neuronal populations. Its trainable dynamics operate over a 2D grid, enabling localized oscillations and structured wavefronts.\n\nReviewer xBBL notes: \"While biologically inspired, the model's structural complexity (2D grids, dynamic interactions) may pose challenges for scaling... The paper moves forward the field of oscillatory models.\" This highlights the conceptual novelty and depth of our work.\n\nOur aim is not to outperform existing SOTA models but to assess whether biologically grounded dynamics, such as locally coupled wave propagation, can be integrated into general-purpose, scan-compatible architectures. These mechanisms, rooted in cortical computation, offer viable inductive biases for sequence modeling. While we incorporate SSM-compatible components for benchmarking and interoperability, these are implementation choices rather than conceptual foundations. The hidden state design, spatial organization, and theoretical derivation of BioOSS were independently developed and are fundamentally distinct from prior work, such as LinOSS."}, {"review_id": "gZUnLqnexD", "reviewer": "Reviewer_rG4F", "summary": "This paper proposes a new model inspired by current oscillatory sequence models as well as neuroscience findings. It can be described using coupled PDEs, which can be solved efficiently in the Fourier domain, where the state evolution is linear and associative scans are applicable (which have $\\log T$ time complexity). It reaches SOA performance on multiple tasks.", "questions": "See above", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Results and Significance**\n\n\n**Strenghts**: The results generally appear solid enough. On an appropriately chosen benchmark, the model outperforms existing architectures with comparable amounts of parameters and compute.\n\n**Weaknesses**: The authors write that the model is very hyper parameter sensitive (line 421). It is unclear how much time was spent hyper-parameter tuning relative to the baseline models. In what fidelity was the learning rate (between 0.000001 and 0.005) explored?\n\n\n**Clarity and originality**:\n\n\n**Strengths**: The introduction is well written and nicely motivates the architecture \n\n\n**Weaknesses**: The writing can be improved in two points, namely:\n\n*Comparison to previous work.* I think an important omission in the relation to previous work is from TA Keller & M Welling (on e.g., neural wave machines / traveling waves) which use a very similar setup of neuro-inspired oscillatory RNNs with spatial interactions (the latter being efficiently implemented with convolutions).\n\n*Section 3.2 to 3.5 can be improved significantly in my opinion.* First of all it contains some elaborate details that can be moved to the appendix, for instance section 3.2 is an almost page long derivation, just to end up with a well known euler based integration scheme. On the other hand some essential details (even just a brief description), such as how the PDEs are used in an ML architecture (does the model contain multiple layers of these PDS with MLPs in between?) are omitted. Where/when do we transform back from Fourier domain to time domain?\n\n**Minor**\n\n- There are some inconsistencies in whether $o$ and $p$ are subscripts or superscripts of $k$ (and sometimes they are missing at all).\n- I don´t understand line 114/115, how does the time-dependancy of the input play a role (or how is it addressed later?). \n- Line 118: isn’t this already diagonal by construction, why do we need to the intermediate block-diagonal step.\n- Line 134, you have not explained why you want this diagonal matrix, maybe add something along the lines of “... so that we can perform the scan operation on the diagonalised transition matrix, similar to in Orvieto et al. 2023” \n- Line 178. I don’t understand why this is not $f =  \\frac{\\theta}{ 2 \\pi dt}$.  A 2 dimensional linear system with c.c. eigenvalues with phase $\\theta$ results in a sine and cosine trajectory with $f =  \\frac{\\theta}{ 2 \\pi dt}$, no?", "was_revised": true, "final_justification": "The contribution seems sounds and has reasonable empirical results. The paper needs a bit of rewriting, which I trust the authors will do for the final version.", "rebuttal": "We thank the reviewer for the detailed and constructive feedback. Below, we address each major point.\n\n## 1. Related Work: Keller & Welling (Neural Wave Machines)\n\nDue to space constraints, we focused the Related Work section on LinOSS and its foundational works. A similar concern was raised by another reviewer, and we plan to address this in the camera-ready version by expanding the discussion. The revised paragraphs below reflect our planned addition:\n\n> Recent works have explored oscillatory and wave-like computation in neural systems, but differ significantly from BioOSS in their underlying mechanisms, biological grounding, and task domains.\n> \n> Mechanistically, these models rely on matrix- or graph-based abstractions—such as masked recurrent matrices (e.g., Neural Wave Machines) or spectral graph filters (e.g., Balkenhol et al., 2024)—to simulate wave-like behavior. Physical implementations like Wave-RNNs (Hughes et al., 2019) and water-wave reservoirs (Maksymov, 2023) approximate wave propagation via analog substrates but lack a direct connection to physical wave equations.\n> \n> In terms of biological grounding, many prior oscillatory models, such as coRNN (Rusch & Mishra, 2021), Neural Wave Machines (Keller & Welling, 2023), and Wave-RNNs, are built using generic RNN modules or abstracted couplings with no direct correspondence to cortical architecture. In contrast, BioOSS introduces signal-bearing $p$ and intermediate modulatory $o$ units, inspired by prefrontal microcircuits (Watakabe et al., 2023). Their interaction enables locally modulated wave propagation and a biologically plausible spatiotemporal abstraction.\n> \n> Task-wise, most existing models are evaluated on static visual tasks, such as MNIST or image segmentation, which lack meaningful temporal dynamics (e.g., Liboni et al., 2023; Keller et al., 2024). BioOSS, in contrast, is explicitly designed for spatiotemporal sequence modeling and is evaluated on tasks that require long-range memory, frequency-based recognition, and dynamic signal tracking. Moreover, many oscillatory models, aside from LinOSS, do not provide a principled discretization scheme, making them difficult to integrate into structured state space modeling (SSM) frameworks. BioOSS is a discretized dynamical system with efficient Fourier-domain recurrence, compatible with modern SSMs.\n\nNWM (Keller & Welling) targets visual tasks using recurrent spatially coupled architectures. While related, it is inspired by CoRNN and was not integrated into the common SSM architecture. However, we did experiment with NWM and Welling's subsequent models, applying them alongside BioOSS on visual and auditory datasets. Our FDTD2D structure, emulating wave propagation in brain-like media, yields richer oscillatory hidden states. As a result, BioOSS outperforms NWM and WaveRNN on tasks with strong intrinsic resonance, as shown in Table 1.\n\n**Table 1. Classification accuracy (%) on sMNIST, nsCIFAR-10, and ESC50-10 ms.** Comparison of **BioOSS** with other oscillatory models. The sMNIST dataset involves long-range dependencies in sequential image pixels, nsCIFAR-10 is a noisy sequential variant of the CIFAR-10 image classification task, and ESC50-10 ms is derived from ESC-50 by splitting 5-second clips into 500 × 10 ms frames.\n\n| *Model*        | *sMNIST*   | *nsCIFAR-10*   | *ESC50-10 ms* |\n|----------------|------------|----------------|----------------|\n| NWM            | 98.6       | 56.2           | 6.8            |\n| coRNN          | 99.3       | 59.0           | 20.8           |\n| WaveRNN           | 97.6       | 55.0           | 18.8           |\n| **BioOSS**     | 98.1       | 54.2           | 35.0           |\n\n## 2. Section 3.2: Explicit Discretization Justification\n\nIn the final version, we will revise this section to better motivate why Section 3.2 includes a detailed derivation, rather than simply stating the use of a common Euler discretization. The key point is that standard formulations often use Euler updates implicitly, which prevents the construction of an explicit coupling matrix. This lack of explicit structure hinders the ability to apply backpropagation through the spatially coupled PDE system.\n\nOur formulation makes the coupling matrix fully explicit and differentiable, which is crucial for enabling backpropagation in the PDE-based dynamics. We will move the derivation to the appendix and keep a concise version in the main text. We will also include a full algorithmic description of the update steps in the appendix to support reproducibility.\n\n## 3. Architecture Clarification and Algorithm Update\n\nWe will revise Figure 1 to show the full model, including input/output layers, to better communicate how the PDE dynamics are embedded in the ML pipeline. While a pseudocode version is included in Appendix C (Algorithm 1), it lacks sufficient detail for the hidden layer's spatial updates. We will update it to the following form in the final version:\n\n**Algorithm 1: Training BioOSS with Spatio-Temporal Layer**\n\n```\nInput: Input sequence u  \nOutput: Output sequence s after L layers\n\nfor l = 1 to L do  \n    // Step 1: Spatio-temporal PDE-based propagation  \n    ∇p ← SpatialGradient(p)  \n    ∇·o ← SpatialDivergence(o)  \n    o* ← o - Δt·∇p  \n    p* ← p - Δt·(c²⊙∇·o) + Δt·B·u\n    o ← o* / (1+Δt·ko)  \n    p ← p* / (1+Δt·kp)  \n    x ← concat(p, o)  \n    \n    // Step 2: Linear projection and residual update  \n    z ← Wz · x  \n    gate ← σ(Wg · x)  \n    x_out ← gate ⊙ tanh(z) + (1 - gate) ⊙ x_prev  \n    \n    // Step 3: Readout  \n    y ← C · x_out + D · u  \n    y ← GELU(y)  \n    u ← GLU(y) + u_prev  \nend for\n\nReturn final output s ← W_out · y + b_out\n```\n\n## 4. Clarification on Fourier Domain Use and Eigenstructure\n\nThe eigenstructure is used solely for theoretical analysis. Specifically, the matrix Λ contains the complex eigenvalues (mapped to frequency components), and P contains their associated eigenvectors. Though training is time-domain, we use $A = P \\Lambda P^{-1}$ to:\n\n- Analyze and visualize the frequency-selective behavior of each node.\n- Justify the scan-efficient computation when approximated via block-wise diagonalization.\n\nWe will make this transformation and its purpose more explicit in Section 3.3.\n\n## 5. Notation and Specific Line Fixes\n\n- **Subscript/superscript consistency:** Inconsistencies (e.g., $k_p$, $k_o$) will be corrected throughout the manuscript.\n\n- **Line 114–115:**  \nThe input’s time-dependency drives internal state changes in BioOSS via the term $M^{\\text{Damp}^{-1}} F^{\\text{Velocity}}$, which injects external stimuli at each step. While $M^{\\text{Damp}^{-1}} M^{\\text{Velocity}}$ governs spatial wave propagation—analogous to a reservoir—the input acts like a stone, triggering dynamic oscillations. The phrase “not effective” refers to **computational inefficiency**. To enable parallel scan operations, we extract local eigenvalues to construct a diagonalized transition matrix.  \n**Revision:**  \nWe will revise the sentences as follows:  \n> \"Although $M^{\\text{Damp}^{-1}} M^{\\text{Velocity}}$ models internal propagation, it does not enable efficient temporal processing of input. Therefore, we extract local eigenvalues to construct a diagonal system matrix for parallel computation.\"\n\n- **Line 118–121:**  \nTo clarify the inversion step in Line 118: since $M^{\\text{Damp}}$ is diagonal by construction, its inverse can be directly computed by taking the reciprocal of the diagonal entries. The current explanation introduces unnecessary detail, assuming a more general block-diagonal case, which will be removed in future revisions.\n\n- **Line 134:**  \nThe need for diagonalization via eigendecomposition $A = P \\Lambda P^{-1}$ was not sufficiently motivated. The original transition matrix lacks associativity, limiting parallel computation. Inspired by Orvieto et al. (2023), we adopt eigendecomposition to obtain a diagonal matrix $\\Lambda$, enabling scan operations for efficient recurrent updates and reducing the computational complexity from $\\mathcal{O}(N)$ to $\\mathcal{O}(\\log N)$.  \n**Revision:**  \nWe will add the following clarification:  \n> \"Following SSM literature (e.g., Orvieto et al., 2023, Rush et al., 2024), we apply eigendecomposition $A = P \\Lambda P^{-1}$, where $\\Lambda$ is a diagonal matrix. This diagonalization allows the use of scan operations across time steps, significantly reducing the computational cost of sequential updates.\"\n\n## 6. Frequency Mapping (Line 178)\n\nFrom Equation 16, eigenvalues λ₂,₃ form complex conjugate pairs. When mapped to the unit circle, they symmetrically align around the real axis. Therefore, the angular frequency associated with one eigenvalue corresponds to half a revolution over a sampling period, leading to the frequency estimate:\n\nf = θ/(π Δt)\n\ninstead of θ/(2π Δt), as would be used in continuous-time systems. This formulation is consistent with our discrete-time eigenvalue interpretation.\n\n## 7. Sensitivity of Hyperparameter\n\nBioOSS is more sensitive to hyperparameters—particularly the learning rate—due to its complex internal structure. The model incorporates nonlinear, spatially localized connectivity through trainable damping and propagation dynamics, resulting in a highly non-smooth optimization landscape. As a result, suitable learning rates vary significantly across datasets and are typically much smaller than those used in other SSMs. This makes tuning more time-consuming and sensitive compared to standard models.\n\n## 8. Section Organization\n\nWe will move the “Limitation” section to the appendix."}, {"review_id": "DSOFV589zq", "reviewer": "Reviewer_p2sQ", "summary": "The authors present a locally coupled oscillatory state space model with two populations of coupled neurons, governed by coupled PDEs. The authors evaluate the proposed model in Fourier space to allow for efficient recurrent computation through the well-known parallel associative scan operation. They evaluate the model's empirical performance on traditional sequence modeling tasks and demonstrate slightly improved performance over recent state of the art.", "questions": "- Can the authors expand on the connection of their model to the results of Watakabe et al 2023? From my reading of the paper, it is not precisely clear how their specific choice of P and O neurons has much relation to these results which instead describe diffuse/patchy connections based on their associated targets. The presented BioOSS model does not seem to recapitulate these two types of projections. Is the idea that the diffuse connectivity is the 'modulation'? And the columns are the P neurons? Where are the patchy connections in this model?\n\n- Section 3.3 is too detailed for the main text and should largely be moved to the appendix in my opinion, with greater intuition for the proposed model and the resulting final equations in the main text. \n\n- Can the authors justify their choice of the particular dynamical system more explicitly? Either from the biological point of view, directly connecting it to natural observations (with explicit abstractions), or through ablation? For example, are the O neurons really necessary in this setup? What do they add that would be different from the Neural Wave Machine type of architecture, or LinOSS with local connectivity, for example? \n\n- Section 4.2 seems bizarrely out of place. What is the take-away from this analysis? It is clear that a dynamical system of oscillations will be dependant on the initalization, and a network of locally coupled oscillators will be dependant on connectivity eigenmode initalization. However this initalization does not appear relevant to any of the following experiments, and therefore seems to lack purpose in the text. I would suggest authors move this to the appendix in favor of including more related work, ablations, and justification for the model. \n\n- How were train-test splits for 4.3.1 chosen? This suite of tasks is known to be very specific to the choice of train-test split (from my own extensive experience with this data set).\n\n- Section 4.3.2 text is largely repetitive of the tables in the section. There is no need to repeat the exact numbers from the table in the text. This can be removed to make more space for substantive material as discussed above.", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The study of spatiotemporal computation in a computationally efficient manner is very welcomed and the proposed methodology appears sound. It is encouraging that the authors appear to have gotten this proposed model to work in the existing SSM context given that it is quite a complex model. \n- The inclusion of different types of neurons inspired by Watakabe et al. 2023 is an interesting novelty and improves the works originality and significance. \n- The stability analysis of Section 3.4 is a nice addition and appreciated. \n- The experimental results are promising although the benchmarks used are all small scale and known to be noisy.\n\n**Weaknesses** \n- Related work section is extremely short and missing critical related work. Specifically the work of Keller & Welling (2023) (see citation below) explicitly builds a recurrent locally connected network of coupled oscillators for performing sequence modeling, also inspired by cortical spatiotemporal dynamics. Furthermore, this is far from the first work to build a computational model which leverages such dynamics, and the authors should acknowledge this in the paper and further describe how their current work differs. Without this comparison, the novelty and originality of their approach is very hard to discern. A non-exhaustive list of papers that the authors should both read and cite is listed below:\n      - Hughes, T. W., Williamson, I. A., Minkov, M., & Fan, S. (2019). Wave physics as an analog recurrent neural network. Science advances, 5(12).\n      - Rusch, T. K., & Mishra, S. (2021). Coupled oscillatory recurrent neural network (coRNN): An accurate and (gradient) stable architecture for learning long time dependencies. arXiv preprint arXiv:2010.00951. Retrieved from https://arxiv.org/abs/2010.00951.\n\t- Keller, T. A., & Welling, M. (2023). Neural wave machines: Learning spatiotemporally structured representations with locally coupled oscillatory recurrent neural networks. In Proceedings of the 40th international conference on machine learning. PMLR.\n\t-  Keller, T. A., Muller, L., Sejnowski, T., & Welling, M. (2024). Traveling waves encode the recent past and enhance sequence learning. In The twelfth international conference on learning representations. Retrieved from https:// openreview.net/forum?id=p4S5Z6Sah4\n\t- Balkenhol, J., Händel, B., Biswas, S., Grohmann, J., v. Kistowski, J., Prada, J., . . . Dandekar, T. (2024). Beyond- local neural information processing in neuronal networks. Computational and Structural Biotechnology Journal, 23, 4288-4305. Retrieved from https://www.sciencedirect.com/science/article/pii/S200103702400360X doi: https://doi.org/10.1016/j.csbj.2024.10.040\n\t- Jacobs, M., Budzinski, R. C., Muller, L., Ba, D., & Keller, T. A. (2025). Traveling waves integrate spatial information through time. arXiv preprint arXiv:2502.06034. Retrieved from [https://arxiv.org/abs/2502.06034](https://arxiv.org/abs/2502.06034).\n\t- Maksymov, I. S. (2023). Analogue and physical reservoir computing using water waves. Retrieved from https:// arxiv.org/abs/2306.09095\n\t- Liboni, L. H. B., Budzinski, R. C., Busch, A. N., Löwe, S., Keller, T. A., Welling, M., & Muller, L. E. (2023, November). Image segmentation with traveling waves in an ex- actly solvable recurrent neural network. arXiv. Retrieved 2024-11-19, from http://arxiv.org/abs/2311.16943 (arXiv:2311.16943)\n\t- Effenberger, F., Carvalho, P., Dubinin, I., & Singer, W. (2025). The functional role of oscillatory dynamics in neocortical circuits: A computational perspective. Proceedings of the National Academy of Sciences, 122(4), e2412830122. Retrieved from https://www.pnas.org/ doi/abs/10.1073/pnas.2412830122 doi: 10.1073/pnas .2412830122\n\t- Even more work in the neural field theory literature... \n- Because of a lack of related work and ablations, it is not clear how different the proposed model is different from prior work. For example, it is not clear how the proposed model differs from an accelerated version (through parallel scan) of the Neural Wave Machine work. Similarly, it is not clear how the model differs from a locally-connected version of the LinOSS model. (Clearly the proposed model has an additional type of neuron with different dynamics, but the inclusion of this additional complexity is unjustified without results showing that it improves performance over the simpler alternative.)\n- The definition and associated derivation of the model in section 3 is very hard to follow, despite the reviewer having experience with wave equations, and SSMs. Specifically, the space over which the neurons are defined should be stated before beginning with the derivation, and it should be made explicitly clear when switching from continuous to discrete spatial dimensions. \n    - Relatedly, equations 1 and 2 are a bit confusing because the image in figure 1 is defined over a discrete grid, while the equations appear to be implicitly defined over continuous space. These should be made explicit.\n- There are no ablations for the proposed architectural components (such as the choice of two types of neurons), and the justification for the modeling choice is very limited (with really only a single reference to Watakabe 2023 for motivation). \n- Despite the authors claiming that their model supports traveling waves, there is no analysis or visualization which demonstrates this. The Figures in the main text and appendix depict what looks like standing waves in the corresponding frequency bands. \n- In the paper the authors write: \"This paper primarily focuses on the exploration of oscillatory-based spatio-temporal model and sequential tasks. Therefore, we do not include Transformer models, as they are end-to-end frameworks\". \n\t- The use of the phrase 'end-to-end' here doesnt make sense. All of these models are end-to-end, and transformers are also used as sequence models, as the authors admit in the introduction, so the choice not to compare with Transformers appears more biased towards making their model look better rather than a true methodological difference.", "was_revised": true, "final_justification": "The paper presents an interesting method for spatiotemporal computing that fits within modern state space model frameworks for efficient training and inference. It has novelty in the use of two coupled populations of neurons, and the demonstrated results are promising. The related work is significantly lacking, but the authors have promised to fix this. They have explored additional ablations which improved the work.", "rebuttal": "We thank the reviewer for their constructive feedback. We address the main concerns and clarify key distinctions with prior literature.\n\n## Major Concern 1: Missing Related Work and Comparison to Similar Models\n\nWe acknowledge the need for a more comprehensive discussion of prior works. Due to space limits, our original submission focused on comparing to LinOSS, the most structurally similar SSM model. In the final version, we will expand the Related Work section to clarify how BioOSS differs in **mechanism**, **biology**, and **tasks**. As reviewer rG4F raised the same issue, we will expand the Related Work section using the revisions outlined in our response to rG4F.\n\n## Major Concern 2: Lack of Ablation Studies and Justification for Architectural Components\n\n**Reviewer:** It is unclear what the contribution of the $o$ neurons is, and how the proposed model differs from LinOSS + local coupling or NWM. No ablations are shown.\n\n**Response:** \n\n### Clarifying the Contribution of $o$ Neurons\n\n$O$ neurons in **BioOSS** modulate the spatiotemporal dynamics of $p$ neurons, which propagate signals. This two-field setup is inspired by the FDTD method, where interleaved state variables (e.g., pressure and velocity) enable wave propagation. Analogously, BioOSS separates signal transmission ($p$ neurons) from modulation ($o$ neurons), enabling control over wave speed and damping across the 2D grid.\n\nBiologically, this aligns with cortical circuits, where modulatory neurons (e.g., interneurons, layer-specific cells) influence gain and oscillatory dynamics without directly relaying external stimuli (Watakabe et al., 2023). By enabling $o$ neurons to modulate local propagation kernels, BioOSS supports spatially varying dynamics—capabilities that single-field models like LinOSS lack.\n\n### Distinctions from LinOSS and Neural Wave Machines (NWM)\n\n**BioOSS** differs fundamentally from **LinOSS** and **NWM** in physical formulation, spatial structure, and frequency-selective modulation: LinOSS models second-order dynamics with global or optionally local coupling but lacks spatial structure, physical grounding, and a separation between signal and modulatory components. NWM introduces wave-like behavior via locally coupled oscillatory RNNs on structured grids, capturing cortical-like topography, but it is not based on PDEs and does not distinguish between signal and modulation. In contrast, **BioOSS** is derived from FDTD principles, using two interleaved fields to provide a physically and biologically grounded mechanism for localized propagation and modulation.\n\nMoreover, $p-o$ coupling introduces learnable, frequency-selective resonance at each grid location, inspired by mixed selectivity in neural systems. This allows BioOSS to effectively model oscillatory and structured temporal patterns—an essential feature for rhythm-sensitive tasks.\n\n### On the Inclusion of an Ablation Study\n\nWe agree that ablation studies and component-level justification are essential. Removing $o$ neurons disrupts the core design of BioOSS.\n\nThe two-field structure stems from the FDTD2D formulation, which inherently relies on interleaved fields to simulate wave behavior. $O$ neurons control key parameters such as propagation speed and damping. Removing them breaks the PDE structure and prevents simulation of wave propagation and modulation.\n\nAdditionally, BioOSS is discretized to remain compatible with the **scan** operator used in Structured SSMs. Removing $o$ neurons would necessitate a new recurrence rule incompatible with both FDTD and biological modularity.\n\nTo assess $o$ neurons’ contribution, we implement a simplified variant on shorter sequences and additional tasks:\n\n- **Single Field**: Removes $o$ neurons, reducing the system to a diffusion-like update of the P-field. Input modulation and resonance are disabled.\n\n#### Ablation Results\n\n| Model         | sMNIST         | nsCIFAR-10      | ESC50-10ms      |\n|---------------|----------------|------------------|------------------|\n| Full BioOSS   | **98.1**       | **54.2**         | **35.0**         |\n| Single Field  | 92.26 (-5.84)  | 37.91 (-16.29)   | 29.75 (-5.25)    |\n\nThe performance drop highlights the role of $o$ neurons in enabling spatially variant dynamics and frequency modulation, which single-field systems cannot capture. This supports both the modeling rationale and the empirical effectiveness of the architecture. \n\n## Major Concern 3: Unclear Derivation in Section 3\n\n**Reviewer:** The derivation is difficult to follow.\n\n**Response:** We acknowledge the confusion and will take the following actions:\n\n- Clearly define the spatial domain of the neurons (2D grid) at the beginning of Section 3.\n- Clarify when we move from continuous PDEs (Eq. 1–2) to the discretized system (Eq. 5–9), emphasizing that we work on a finite 2D lattice with fixed Δx.\n- Improve Fig.1 by explicitly marking the grid resolution and denoting the discrete structure alongside the PDE formulation.\n\n## Major Concern 4: No Demonstration of Traveling Waves\n\n**Reviewer:** There is no evidence that BioOSS supports traveling waves; observed dynamics appear to be standing waves.\n\n**Response:** While the visualizations in the current manuscript primarily show stationary patterns, BioOSS is formulated from a wave propagation PDE (via FDTD), which intrinsically supports traveling wave solutions. As such, the hidden layer dynamics inherently include both standing and traveling wave components.\n\nIn the current experiments, we focused on long-duration time-series tasks, where wavefront propagation is less visually salient than in spatially structured visual tasks. Long sequence lengths (several hundred time steps) make clear animations or frame-wise visualizations impractical. We will include a supplementary traveling wave visualization in a 2D setting.\n\n## Major Concern 5: Justification of Not Including Transformer Baselines\n\n**Reviewer:** The justification \"Transformers are end-to-end\" is incorrect. All models are end-to-end.\n\n**Response:** We agree our phrasing was misleading. Our intent was to highlight fundamental architectural differences rather than training paradigms. BioOSS is designed to model local dynamical systems with explicit spatiotemporal continuity, grounded in oscillator-based dynamics. In contrast, standard Transformers rely on perceptron-based components and model global dependencies via attention mechanisms, without encoding spatiotemporal continuity or locality.\n\n## Clarifications on Specific Reviewer Questions\n\n### Q1: Connection to Watakabe et al. (2023) and Biological Motivation\n\n**A1:** In *Watakabe et al.*, two key projection motifs in primate prefrontal cortex (PFC): (1) **patchy, columnar projections**, radially oriented and localized in submillimeter-scale patches, and (2) **diffuse projections**, which spread widely and contribute to gradient-based organization across the cortex. Both projection types are topographically structured, supporting spatially coherent cortical coordination.\n\nOur BioOSS model was developed as a **computational abstraction** of these biological principles:\n\n- **$p$ neurons** in BioOSS emulate the **columnar units** described in Watakabe et al., corresponding to localized clusters of pyramidal neurons, the main excitatory signal carriers. Arranged on a 2D grid, their local recurrence and frequency-specific oscillations capture the notion of **discrete processing modules** akin to patchy projections.\n\n- **$o$ neurons** model the **lateral propagation dynamics**, inspired by **diffuse projections** that modulate activity across spatially distant areas. In our formulation, these $o$ neurons control wave propagation velocities, enabling low-amplitude, spatially distributed modulation that interacts with the localized oscillations of $p$ neurons. This provides a computational analogue to the diffuse projections described by Watakabe et al. (2023).\n\nWhile we do not simulate anatomically resolved axonal arborizations, our model captures the functional consequences of both projection types: localized recurrent activity (patchy/columnar) and diffuse global modulations, jointly driving emergent dynamics. The \"diffuse connectivity as modulation\" interpretation suggested by the reviewer aligns well with our intent for the $o$ neuron population, and we appreciate the clarification opportunity.\n\n### Q2: Justification of the Dynamical System and Role of $o$ Neurons\n\n**A2:** This is addressed in Major Concern 2.\n\n### Q3: Section 3.3 Too Detailed\n\n**A3:** We agree. We will move much of Section 3.3 to the Appendix, retaining a high-level summary with the scan efficiency idea and eigen-decomposition rationale in the main text.\n\n### Q4: Section 4.2 Lacks Relevance to Main Experiments\n\n**A4:** Section 4.2 supports the theoretical foundation of BioOSS by validating its frequency-selective behavior and the eigenvalue frequency mapping in Section 3.5. By initializing spatial regions with distinct eigenmodes and applying identical white noise, we show that BioOSS generates region-specific oscillations consistent with their assigned bands. This confirms the theory and shows the model’s potential for EEG-like simulations, where each region can be selectively initialized. While spatial effects are harder to quantify, later experiments show hidden state frequencies align with or adapt to input signals. These findings demonstrate BioOSS’s capacity for rich oscillatory dynamics and mixed selectivity. We will move the full analysis to the Appendix and briefly explain its relevance in the main text.\n\n### Q5: Clarify Train-Test Splits in UEA Tasks\n\n**A5:** We followed Walker et al. [2024] and Rusch et al. [2024]'s 70:15:15 split. This is now clarified in Section 4.3.\n\n### Q6: Section 4.3.2 Text Is Redundant with Tables\n\n**A6:** Agreed. We will trim this section to avoid restating results and use the space for interpretation and ablation."}]

Author rebuttal:

Meta-review: This paper proposes a biologically inspired oscillatory neural network architecture. Some reviewers appreciated the model formulation and empirical tests, and think that the model represents a step forward for oscillatory networks. The main concern was that the authors do not properly situate their work within the context of the somewhat extensive existing literature on oscillatory models. The authors promised to revise their paper to properly situate their work, which satisfied the reviewers. Provided the authors revise their work as promised, I recommend acceptance.
Decision: Accept (poster)