Patterns of synchronized clusters in adaptive networks

Cluster synchronization (CS)—where nodes form groups with identical within-cluster dynamics—is widespread in natural and engineered networks, from robotics to biological and ecological systems. Many real systems are also adaptive (plastic), with connectivity coevolving with node dynamics. While prior work has emphasized adaptation’s role in promoting full synchronization and presented MSF-based stability analyses for fully synchronized regimes in plastic networks, CS in adaptive networks remains less understood. This paper addresses: (i) which CS patterns can exist under adaptation; (ii) which of these patterns are dynamically selected (i.e., stable attractors) with adaptation; and (iii) whether linear stability can be analyzed in a reduced space to overcome large dimensionality. The authors show that feasible CS patterns are determined by network topology (within a broad class of adaptation rules), their stability and selection depend also on dynamics and adaptation, and that substantial dimensionality reduction is achievable, enabling analysis of multi-stable synchronized activity in adaptive networks with delays and multiple layers.

The study builds on extensive literature on synchronization and cluster synchronization in complex networks, including applications in robotics, brain dynamics, ecological systems, and engineered systems. Prior research has explored adaptive (coevolutionary) networks and the impact of plasticity on synchronization, including MSF analyses for full synchronization in plastic networks. Non-adaptive networks are known to support coexisting CS patterns, but the interplay between CS and adaptation has been less explored, partly due to approximate synchronization in heterogeneous settings. Recent works addressed adaptive coupling rules (e.g., Hebbian, Wang–Rinzel), phase-oscillator networks with adaptive interactions, and methods for identifying equitable partitions and performing simultaneous block diagonalization (SBD) for stability analysis in non-adaptive settings. This paper extends these ideas to adaptive, multi-layer networks with delays, providing a general framework for existence and stability of CS patterns and demonstrating multi-stability induced by adaptation.

General model: An adaptive multi-layer network of N nodes with L connection types (layers) and possible delays is modeled by delay differential equations ẋi(t) = Fi(xi(t)) + Σl Σj α aij^(l) g^(l)(xj(t − τl)) and adaptive couplings ḃij(t) = −ε(bij(t) + H^l(xi(t), xj(t))). The adjacency matrices A^(l) specify layer connectivity; α is coupling strength; g^(l) are coupling functions; ε sets adaptation time scale; H^l defines plasticity (e.g., Hebbian, Wang–Rinzel). Existence of CS: Nodes are partitioned into clusters C1,…,CQ. Any equitable partition for A^(l) yields a flow-invariant cluster-synchronous solution. Equitable partitions (minimum number Q) are found via a coloring algorithm (variation of Hasler–Belykh). Quotient network: For a CS pattern, dynamics reduce to a Q-node quotient system with cluster states sp(t) obeying ṡp = fp(sp) + Σq Σl r(pq)^(l) kl pq g^(l)(sq(t − τl)), and adaptive weights k̇l pq = −ε(kl pq + H^l(sp,sq)), where r(pq)^(l) are entries of the quotient matrices R^(l). Linear stability and dimensional reduction: Linearize about the CS manifold using perturbations wi = xi − sp(i) and yij = bij − kl p(i)q(j). Introduce cluster indicator matrices Ep and assemble variational equations. Apply a canonical transformation T from simultaneous block diagonalization of {A^(l)} and {Ep} to separate longitudinal (along-manifold) and transverse (stability-relevant) modes. In transformed coordinates, transverse perturbations η⊥, ξ⊥ satisfy decoupled block systems with time-varying matrices P1–P4 (given in Methods) and delayed terms. The transverse equations split into M independent diagonal blocks of sizes dα; some entries are identically zero and can be dropped. Stability is evaluated via transverse Lyapunov exponents (TLEs). Complexity reduction: Original system size Norig = mN + L N^2 (for full adaptive matrices) matches the variational system, but after SBD the number of equations to analyze is Nred = (m + LQ)(Q + D), where D is the sum of distinct block sizes across transverse blocks. For all-to-all topology, maximal reduction occurs with scalar blocks: M = N − Q, D = 1, giving Nred = (m + LQ)(Q + 1). Case studies: The framework is applied to (i) a plastic single-layer network of Wilson–Cowan neural masses (global coupling, uniform delay, Hebbian plasticity) to study encoding/decoding of stimuli via CS patterns; (ii) a two-layer (multiplex) adaptive network of phase oscillators modeling tissue (parenchyma/immune) with an age-related plasticity parameter β and a small fraction r of pathological nodes, assessing healthy vs pathological multifrequency CS; and (iii) a small Erdős–Rényi network of chaotic Lorenz oscillators with Hebbian adaptation, verifying TLE-based stability regions vs direct simulations. Throughout, the quotient network and SBD-based MSF approach provide efficient stability assessment and parameter sweeps.

- General results: (i) Feasible CS patterns (both stable and unstable) are determined by network topology within the considered adaptation rules; (ii) which CS patterns are stable (and thus dynamically selected) depends also on node dynamics and adaptation; (iii) the stability analysis can be reduced dramatically in dimension via SBD-based MSF, often by orders of magnitude. - Neural masses (visual memory encoding): A fully connected Wilson–Cowan network with Hebbian plasticity and delay supports stimulus-induced cluster patterns that are always stable (all TLEs negative) for Q = 2 and Q = 3 configurations tested. Stimulating N1 of N nodes yields: (a) no oscillations (low persistent activity); (b) oscillations in the stimulated cluster only; or (c) high/low persistent activity regimes, with boundaries computed semi-analytically. In the oscillatory regime, collective frequencies lie in the β–γ band. Higher global coupling σ reduces the number of stimulated nodes needed for oscillations; the frequency range is robust across σ. For two stimulated groups (Q = 3 overall), clusters C1 and C2 exhibit stable phase-locked β–γ oscillations with phase lag Δ in [π/2, 3π/2], Δ = π when N1 = N2 and deviating otherwise. Large-scale image-based stimulation (N = 40000) shows Δ varies approximately linearly with N1/N in [0.4, 0.6], enabling phase-lag-based population coding of visual patterns. Dimensionality reduction in this experiment lowers the number of equations from N^2 ≈ 1.6×10^6 to Neq = 12 (m = 2, L = 1, Q = 2, D = 1). - Tumor propagation model (two-layer phase oscillators): For N = 200 with small initial pathological fraction r < 0.1 and plasticity phase shift parameter β ∈ [0.40π, 0.55π), a 2-cluster phase-synchronized solution (healthy state) exists and is stable for all tested (β, r) pairs with ⟨ω⟩ ≈ 0. Multi-stability is prominent: splitting healthy nodes into Na initial phase clusters (Na ∈ {6,10,15}) changes outcomes—higher Na expands the pathological region (⟨ω⟩ > 0). Thus, greater initial fragmentation of healthy nodes makes pathology more likely; a single large synchronized cluster is more resilient. Raster plots and coupling snapshots show stronger asymptotic coupling in healthy states than pathological ones. Robustness: introducing heterogeneity ω^(1) = Z(0,Σ) (healthy) and 1 + Z(0,Σ) (pathological) preserves qualitative thresholds for Σ ≤ 10^−2 across Na values. Dimensionality reduction cuts simulation size from Norig = 80400 to Nred = 128 (m = 2, L = 2, Q = 7, D = 1) for Na = 6. - Lorenz network (non-global coupling): In an Erdős–Rényi N = 10, Q = 3 clustering with Hebbian adaptation, TLE maps over (σ, ε) accurately predict cluster stability regions and agree with synchronization error computed from full simulations, while reducing equations from 130 to 36 (m = 3, L = 1, Q = 3, D = 3). Intertwined clusters share TLEs as predicted by block structure. - Overall, the approach consistently enabled accurate stability prediction, identified multi-stable CS patterns, and substantially reduced computation, with maximal reduction for all-to-all topologies (Nred = (m + LQ)(Q + 1)).

The study addresses the core questions on CS under adaptation by showing that topology fixes the admissible CS patterns (via equitable partitions), while dynamics and adaptive rules select which of these become stable attractors. The MSF/SBD framework decouples topology from dynamics, enabling stability analysis in low-dimensional blocks and computation of transverse Lyapunov exponents. Applications demonstrate relevance to neuroscience (population coding via β–γ oscillations and phase lags), pathophysiology (onset of multifrequency pathological clusters controlled by an age-related plasticity parameter and initial clustering), and nonlinear dynamics (chaotic oscillator networks with non-global topology). The findings highlight the prevalence of multi-stability in adaptive networks and the importance of initial conditions (e.g., number and composition of clusters) in determining long-term outcomes. Despite focusing on exact CS and local (linear) stability, the approach provides an effective first-pass tool; robustness to moderate heterogeneity was confirmed in the tumor model through full-network simulations. The dimensional reduction facilitates systematic parameter exploration and identification of stable CS regimes that would be computationally prohibitive otherwise.

The paper introduces a general MSF-based framework for existence and stability analysis of cluster synchronization in adaptive, multi-layer networks with delays, separating topological constraints from dynamical stability and yielding substantial dimensionality reduction. Across three case studies, the method uncovered rich multi-stability and provided mechanistic insights: adaptive neural masses can encode stimuli via stable β–γ oscillatory clusters with phase-lag coding; an age-related plasticity parameter and initial clustering shape healthy vs pathological outcomes in tissue models; and TLE-based stability prediction matches direct simulations in chaotic networks with sparse connectivity. The methodology extends CS analysis to plastic networks with heterogeneous nodes and multiple coupling layers. Future directions include relaxing symmetry requirements, extending to stronger heterogeneity and approximate CS, integrating learning rules for targeted pattern embedding, and applying the framework to additional biological and engineered adaptive systems.

- Symmetry requirements: The method assumes symmetries in coupling structure and strength (e.g., equitable partitions and compatible adaptation), which may be difficult to satisfy in some adaptive systems. - Many distinct node dynamics: When the number of different oscillator types is large (many distinct Fi), the dimensionality reduction benefits diminish. - Local validity: Stability is assessed via linearization (MSF) around exact CS solutions; results are locally valid and may not capture global dynamics or strongly heterogeneous regimes. - Exact CS focus: The framework targets exact cluster synchronization; while some robustness to small heterogeneity is observed, significant heterogeneity precludes exact CS and may require full-network simulations.