Oscillating Topological Neighbourhoods

• Oscillating Topological Neighbourhoods are dynamic constructs where local connectivity patterns periodically fluctuate to enable global reorganization.
• They are pivotal in diverse fields such as statistical physics, neural computation, and metamaterials, offering unified insights into phase transitions and adaptive dynamics.
• Practical models leverage these oscillations to prevent network freezing and achieve topological ergodicity breaking, enhancing the robustness of distributed systems.

An oscillating Topological Neighbourhood (TN) refers to a class of mathematical, physical, or computational constructs where local or global topological properties (neighbourhoods around points, patterns of connectivity, or neighbourhood update ranges in neural networks) fluctuate or periodically reset according to a prescribed scheme. Across domains as diverse as phase transitions in statistical lattice models, dynamics of neural fields, spatial logic, molecular computation, and cognitive modelling, oscillating TNs appear in connection with the propagation, modulation, or organization of nonlocal phenomena that cannot be characterized by conventional local order parameters. The concept is especially rich at the interface of topology, dynamics, and computation, where the spatial or temporal oscillation of a neighbourhood has profound consequences for global behavior, ergodicity breaking, learning capacity, and the emergence of robust patterns.

1. Mathematical Foundations and the Oscillating Topological Neighbourhood in Statistical Lattice Models

The notion of an oscillating topological neighbourhood is rigorously realized in topology-driven phase transitions in constrained lattice models, most notably within the classical monomer-dimer-loop (MDL) model (1504.01566). In this context, the TN represents both the tensor network (TN) data structure encoding all allowed local configurations and the neighbourhood of a state in the high-dimensional configuration space.

The MDL model’s partition function is represented as a two-dimensional tensor network in which each vertex is assigned a local tensor containing information about the presence of monomers, dimers, or loop segments. The phase transition separating the monomer-condensed phase (trivial) from the loop-condensed (topologically ordered) phase does not admit any local order parameter. Instead, nonlocal properties—oscillations between distinct topological sectors in the transfer-matrix spectrum—emerge. In particular, in the loop-condensed phase, the transfer matrix spectrum features two degenerate eigenvalues corresponding to two global topological sectors (even and odd loop parity), and the system is constrained to oscillate within one sector at large scales.

This sectorization produces a form of topological ergodicity breaking: the system's configuration space fragments into “oscillating topological neighbourhoods” defined by parity constraints, between which transitions require global rearrangements, not accessible through local dynamics. These oscillations are manifest in observables such as the nonlocal string order parameter and are rigorously diagnosed by numerical vanishing of the transfer-matrix spectral gap and the emergence of long-range nonlocal order. Thus, oscillating TNs in this setting encapsulate the division of phase space into dynamically disconnected neighbourhoods due to global topological constraints.

2. Oscillating Neighbourhoods in Neural and Cognitive Systems

Oscillating TNs play a foundational role in compute and neurobiologically motivated models of self-organization. In the context of Self-Organising Maps (SOMs), a widely used neural network for topographic mapping, an oscillating TN is implemented by periodically resetting the topological neighbourhood width during training (Revithis et al., 16 Jul 2025). The canonical SOM employs a monotonically shrinking TN width function:

$o(n) = o_0 \cdot \exp(-n/T_1)$

whereby the width decreases throughout training epochs. The oscillating TN introduces a modified function:

$$o'(n) = o_0 \cdot \exp\left( - \frac{[(n+1) \mod t']}{\tau'} \right)$$

By partitioning the training session into intervals of exponential decay followed by resets back to the initial width, this oscillatory behaviour more closely reflects the biological oscillations seen in cortical columnar activity and enables continuous learning and adaptation. This approach maintains the capacity for topological reorganisation throughout map formation and leads to map structures that are nearly equivalent in their global properties to the traditional SOM, given that the cumulative “learning area” (∫ o(n) dn over all epochs) is preserved.

Computationally, oscillating TNs avoid the pathological “freezing” of the network commonly encountered when neighbourhood widths decay too rapidly, facilitating ongoing adaptation. From a cognitive modelling standpoint, they enable more faithful analogues to neural plasticity, with applications demonstrated for modelling neurodevelopmental disorders such as autism (characterized by narrow, unmalleable TNs) and schizophrenia (where premature contraction of the TN leads to pathological persistence of delusional patterns).

3. Oscillatory Behaviour in Topological Dynamical and Logical Systems

The concept of oscillating TNs is closely tied to oscillating sequences in dynamical systems and the logical analysis of spatial neighbourhoods (Fan, 2018, Linker et al., 2020). Oscillating sequences, defined by their vanishing correlations with all polynomial phase functions, are orthogonal to the dynamics of systems with quasi-discrete spectrum. Here, the idea of an oscillating neighbourhood generalizes to the statistical or ergodic behaviour of observables: in a phase space, local oscillatory cancellation properties of the observables mean that when weighted by an oscillating sequence, ergodic averages vanish—even for highly structured polynomial iterates.

In spatial logic, neighbourhood spaces generalize topological spaces by equipping every point with a neighbourhood filter. Bisimulation relations define when two points (or entire neighbourhood spaces) are indistinguishable in terms of spatial logic. The modem notion of "oscillating topological neighbourhood" in this context refers to local regions whose properties (separation, connectedness) may oscillate under logical transformations or temporal dynamics. However, standard spatial logics like SLCS cannot capture such fluctuations, as they are invariant under bisimulations that ignore global topological changes (Linker et al., 2020). This suggests the need for refined logical frameworks to effectively describe oscillatory neighbourhood phenomena.

4. Physical Realizations: Oscillatory Topological Neighbourhoods in Lattice and Photonic Systems

Oscillating TNs are embodied in physical platforms exhibiting time-periodic or spatially oscillatory modulation, yielding novel topological phenomena. In gyroscopic metamaterials, topological neighbourhoods oscillate through lattice deformation or geometric phase modulation, producing edge modes that change chirality or localization as a function of lattice “twist” or interaction strength (Mitchell et al., 2018). In Floquet-engineered waveguide arrays, topological invariants and edge (π) modes oscillate within a driving period, leading to robust edge or corner-localized states and solitons whose profiles recur exactly after each period (Arkhipova et al., 2023). These structures realize dynamically switching or “oscillating” TNs, with anomalous Floquet modes occupying topologically protected neighbourhoods that are accessible or forbidden depending on the phase within each period.

Parametric oscillatory backgrounds in spacetime fields can also drive oscillating TNs at the field level: a time-dependent metric induces parametric resonance in a symmetry-broken field, generating oscillatory excitation and the nucleation of topological vortices and defect lattices (Dave et al., 2019). The spatial arrangement and robustness of such defect neighbourhoods are controlled not only by static symmetry but by the frequency and amplitude of oscillating external fields.

5. Oscillating Topological Neighbourhoods in Neural Data and Persistent Homology

The interplay of neural oscillations and topology is strikingly exemplified in studies of grid cell activity in the mammalian brain (Sarra et al., 31 Jan 2025). Population activity can be shown, via persistent homology, to reside on a toroidal manifold—a topological neighbourhood stabilized by oscillatory modulation in the theta and eta bands (∼8 Hz and ∼4 Hz). Small temporal jitters of spike times, less than these bands' periods (100–500 ms), preserve the toroidal structure, while larger jitters erase it, although hexagonal spatial order remains intact.

The degree of toroidality is quantitatively measured by comparing the normalized bottleneck distance between the observed persistent homology barcode and that of an ideal torus. This provides a meaningful scale-free metric:

$$\Gamma_d(\tau_d,\tau^{ref}_d) = 1 - \widehat{d}_B (\tau_d, \tau^{ref}_d)$$

Oscillatory modulation thus carves out and maintains an “oscillating topological neighbourhood” in high-dimensional neural activity, organizing the population state into an effectively low-dimensional, persistent structure that is robust to noise up to a critical oscillation-dependent threshold. The strength of specific oscillatory components (e.g., the eta/theta power ratio) directly correlates with the resilience of the topological neighbourhood. This suggests that oscillating TNs serve as global organizing principles in real neural systems, beyond what can be inferred from local or pairwise arrangements.

6. Oscillating Topological Neighbourhoods in Computational and Communication Frameworks

Recent advances in topological neural networks (TNNs) illustrate how concepts of oscillating TNs inform network design and analysis (Kouritzin et al., 2023, Fiorellino et al., 14 Feb 2025, Verma et al., 5 Jun 2024). In these models, data is structured over topological spaces (simplicial or cell complexes), and learning propagates over both lower and upper neighbourhoods—capturing higher-order, oscillatory message-passing patterns. For distributed systems implemented over wireless channels, the “oscillation” between lower and upper neighbourhoods is harnessed to enhance robustness to signal fading and noise, yielding architectures that outperform purely graph-based (pairwise) models. The alternating or oscillatory information exchange between topological neighbourhoods enables richer, adaptive representations and decentralized computation.

From a universal approximation perspective, TNs built over Tychonoff spaces (completely regular, Hausdorff) employ families of separating continuous functions whose oscillations over the space ensure the capacity to approximate any uniformly continuous function (Kouritzin et al., 2023). This framework ties together classical neural network theory, deep sets, and stochastic process estimation under the umbrella of oscillatory sensitivity to topological neighbourhoods.

7. Summary and Broader Implications

Oscillating Topological Neighbourhoods (TN) encapsulate a structural and dynamic theme that spans statistical mechanics, dynamical systems, spatial logic, neuroscience, computational intelligence, and physical systems engineering. Their defining feature is the periodic or fluctuation-induced modulation—whether of neighbourhood structure, topological sector, or neighbourhood update rule—that organizes or fragments the global behavior of the system. In all these settings, oscillating TNs are pivotal for (i) accommodating robust nonlocal order in the absence of local markers, (ii) sustaining adaptive plasticity and learning, (iii) supporting distributed computation and resilience, and (iv) mediating topological phase transitions and persistence phenomena. Their analysis and application rely on mathematical tools such as tensor networks, persistent homology, nonlocal order parameters, and decentralized algorithms over complex topologies, and their paper continues to reveal new vistas at the intersection of topology, dynamics, and computation.