Functional control of oscillator networks

The study addresses how to control and prescribe precise patterns of synchrony (functional patterns) in oscillator networks by tuning structural parameters (couplings and natural frequencies). Motivated by the prevalence of oscillatory coordination in natural and engineered systems (e.g., brains, power grids, biological collectives), the authors formulate the research question: can one map structural constraints to exact pairwise functional relationships and design network parameters to enforce desired patterns? They model systems as Kuramoto oscillator networks (allowing cooperative and, where relevant, competitive couplings), define a functional pattern capturing time-averaged pairwise phase correlations, and focus on phase-locked behaviors that underlie many real systems. The importance lies in moving beyond macroscopic synchrony measures to prescribe microscopic pairwise relations, with applications demonstrated in human brain functional connectivity and power grid power-flow redistribution.

Prior work has extensively characterized synchronization phenomena in oscillator networks and control of macroscopic synchrony measures. The most related study tailored interconnection weights and natural frequencies to bound phase cohesiveness, but did not prescribe exact pairwise differences or analyze stability of specific patterns. Classical results address global order parameters and average synchronization levels; cluster synchronization studies enforce coherence within groups, not precise inter-cluster relations. The present work advances by prescribing exact pairwise correlations across all oscillator pairs, providing feasibility and stability conditions, enabling multiple concurrent equilibria, and framing control as convex optimization. It connects to algebraic graph theory (incidence matrices, kernels, M-matrices), structural balance in signed networks, and applications in neuroscience and power systems.

• Modeling: The network consists of n Kuramoto oscillators with dynamics θ̇_i = ω_i + Σ_j A_ij sin(θ_j − θ_i) on an undirected graph, allowing either both positive and negative couplings or only nonnegative couplings. The functional pattern R has entries ρ_ij = ⟨cos(θ_i(t) − θ_j(t))⟩, focusing on phase-locked regimes where phase differences are constant.
• Vector formulation: Using the oriented incidence matrix B and the diagonal matrix D(x) of sines of edge phase differences, the phase-locked condition becomes B D(x) δ = ω, where δ stacks edge weights and ω is the vector of natural frequencies centered to zero mean. Any functional pattern has n−1 degrees of freedom, so desired patterns are specified by n−1 independent phase differences x_desired.
• Feasibility with nonnegative couplings: The existence of δ ≥ 0 solving B D(x) δ = ω depends only on the signs of sin(x) (i.e., sign of D(x)). Feasibility is equivalent to ω lying in the cone generated by the columns of B sign(D(x)). Sufficient algebraic conditions are provided via a subset S of columns ensuring: (i) D_{S,S} B_S^T B_S D_{S,S} is an M-matrix (nonpositive off-diagonals, positive principal minors), (ii) positive projections ω^T B_i D_{ii}(x) > 0 for i in S, and (iii) ω ∈ Im(B_S). Under these, a nonnegative solution exists and is constructed via Moore–Penrose pseudoinverses.
• Strictly positive couplings: A sufficient graph-theoretic condition for δ > 0 is the presence of a directed Hamiltonian path H in the underlying orientation with ω projecting positively on its columns. Then positive weights exist (and can be constructed) that realize the pattern.
• Compatibility of multiple patterns: For fixed nonzero weights 𝔅, all compatible equilibria x^(i) satisfy sin(x^(i)) ∈ diag(𝔅)^{-1}(𝔅 ω + ker(𝔅)), linking multiplicity of patterns to the kernel of the incidence matrix. Trees (ker=0) admit finitely many patterns (phase differences 0 or π); cycles (ker spanned by all-ones) and complete graphs admit infinitely many families (including splay states), parameterizable along directions in ker(B).
• Stability analysis: The Jacobian at a phase-locked pattern is J = −B diag(A_ij cos x_ij) B^T, a Laplacian scaled by cosines. If all |x_ij| < π/2, J is stable. With both positive and negative allowed couplings, one can choose signs of A_ij to align with cos(x_ij) and ensure a positive Laplacian and stability. In positive-only networks, entries with cos(x_ij) < 0 yield a signed Laplacian; structural balance of the cosine-scaled network implies instability. For line and cycle graphs, simple instability/stability conditions are derived, including limits on the number and size of phase differences exceeding π/2.
• Convex design problems: Control is formulated as convex optimization. Given initial weights δ and target x, adjust weights α to minimize ||α||_2 subject to B D(x) (δ + α) = ω and δ + α ≥ 0. For multiple target patterns, enforce constraints for all patterns. To promote stability in positive networks when some cos(x_ij) < 0, a heuristic convex program minimizes the corrections on edges with negative cosine contributions, leveraging Gershgorin disks to shift eigenvalues left. With unconstrained parameters, a joint program adjusts both weights and natural frequencies; a closed-form solution for frequency corrections β* is provided to satisfy the linear constraint for any chosen weights.
• Applications: Procedures to extract phases from fMRI time series, identify phase-locked windows, compute functional patterns, infer ω from desired x, and simulate Kuramoto dynamics on empirical brain structural networks. In power systems, mapping of a structure-preserving grid model to Kuramoto parameters (A_ij from admittances and voltages, ω_i from active powers and damping), simulation of faults, and recovery of pre-fault power flows via minimal local adjustments to line parameters using the convex programs.

• Feasibility mapping: Achievability of a functional pattern with nonnegative weights reduces to the cone condition on ω relative to columns of B sign(D(x)). Networks can realize a continuum of patterns sharing the same sign pattern of sin(x).
• Strict positivity via Hamiltonian paths: If the graph contains a directed Hamiltonian path with ω positively projecting on its incidence columns, then strictly positive weights exist that realize the desired pattern.
• Multiple compatible patterns: The set of coexisting equilibria is characterized by sin(x) lying in an affine subspace determined by diag(𝔅)^{-1}(𝔅 ω + ker(𝔅)). Trees admit 2^{n−1} binary patterns; cycles and complete graphs admit infinite families including splay states and other parameterized equilibria.
• Stability conditions: If all target phase differences satisfy |x_ij| < π/2, the Jacobian is a positive Laplacian and stable. In networks constrained to positive weights, any structurally balanced cosine-weighted network implies instability of the corresponding pattern. For line and cycle networks, only limited numbers/magnitudes of phase differences exceeding π/2 can be stable; for cycles with positive weights, stability is only possible if at most one difference exceeds π/2 and remains below an approximate bound (~1.79 rad).
• Convex control: The weight-tuning problem to realize one or multiple patterns is convex and efficiently solvable; different weight configurations can yield the same pattern. A Gershgorin-guided heuristic convex program reduces destabilizing edges (where cos(x_ij) < 0), shifting eigenvalues left and promoting stability. With unconstrained parameters, a closed-form frequency correction ensures feasibility for any weight choice.
• Brain application: Using an empirically reconstructed structural network (cingulo-opercular system, n=12), phases extracted from filtered fMRI (0.04–0.07 Hz) yield desired x; setting ω = B D(x) reproduces functional connectivity with small error (example window: ||R − F||_2 = 0.2879; generally ||F − R||_2 ≤ 1), and Jacobian analysis predicts stability consistent with simulations.
• Power grid application: In the IEEE 39-bus New England system, after a line disconnection fault, solving the convex program with an ℓ1 objective identifies minimal, local parameter changes (e.g., neighboring line impedances) that recover pre-fault active power flows while maintaining synchronization. The approach extends to more detailed generator and lossy line models.

The framework establishes a principled, interpretable mapping from structural parameters (weights and natural frequencies) to pairwise functional relationships in oscillator networks, directly addressing the challenge of prescribing microscopic synchrony patterns. By casting control as convex optimization with provable feasibility conditions based on incidence geometry and graph paths, the method enables both single and multiple target patterns and reveals how network structure (e.g., density, presence of Hamiltonian paths) and nodal heterogeneity (ω) shape functional capabilities. Stability analysis delineates when patterns are robust, especially straightforward when both positive and negative interactions are allowed, and provides a practical heuristic to promote stability in positive-only networks. Applications to human brain activity suggest that regional metabolic changes (natural frequencies) can drive shifts among observed functional patterns over a fixed structural backbone, while in power grids, minimal local tuning can restore desired power flows post-fault and avoid adverse phenomena such as Braess’s paradox. Overall, the results connect structure, function, and control in oscillator networks, expanding beyond macroscopic synchronization metrics to precise pairwise control.

The study introduces a general, computationally efficient framework to design and control functional patterns in oscillator networks by optimally tuning interconnection weights and natural frequencies. It provides algebraic and graph-theoretic feasibility conditions (including for strictly positive designs), characterizes coexistence of multiple patterns via the kernel of the incidence matrix, and analyzes stability, with a practical heuristic for positive-only networks. Convex optimization formulations enable single and multi-pattern assignment, with extensions to unconstrained parameter tuning and directed networks. Demonstrations in brain and power networks showcase practical utility. Future directions include deriving general stability conditions for positive networks, systematically characterizing network structures that allow multiple prescribed equilibria (relevant to memory systems), incorporating sparsity and access constraints, modeling with higher-dimensional oscillators for task-related brain activity, handling partial synchronization regimes, and integrating system identification when network parameters are unknown.

• Modeling scope: Kuramoto phase-only dynamics neglect amplitude effects; higher-dimensional oscillators may be needed for certain applications (e.g., task-evoked brain activity).
• Phase-locked focus: Methods are developed for phase-locked configurations and may not capture regimes with partial or transient synchronization.
• Stability in positive networks: General stability guarantees for patterns with some |x_ij| > π/2 and positive-only couplings remain challenging; proposed procedure is heuristic.
• Basins of attraction: The basin sizes of target equilibria are not analytically characterized; estimating basins in large heterogeneous Kuramoto networks is difficult.
• Parameter knowledge: In scenarios with incomplete or uncertain network parameters, additional identification steps are required.
• Multiplicity of equilibria: The number of equilibria grows with network size, complicating global guarantees and initialization strategies.