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Synchronization of Interconnected Systems with Applications to Biochemical Networks: an Input-Output Approach
L. Scardovi, M. Arcak, E. D. Sontag
TL;DR
The paper addresses synchronization conditions for nonlinear networks whose compartments contain internally interconnected species and are coupled across compartments. It combines input-output subsystem properties with network structure, obtaining synchronization results for external inputs and state-space systems. Under relaxed-cocoercivity, connectivity, and diagonal-stability conditions, output disagreement is controlled by input disagreement, while suitable state-space assumptions yield asymptotic synchronization toward the isolated system’s limit set.
Problem
The paper studies how to characterize synchronization in nonlinear networks with internal species interactions and external coupling, motivated by cellular signaling networks.
Method
The paper combines input-output properties of species operators with interconnection structure, using relaxed cocoercivity, graph connectivity, and dissipativity-matrix diagonal stability.
Results
The theory relates small external-input disagreement to small output disagreement and extends to state-space models with asymptotic synchronization under reachability and related assumptions.
Takeaways & Limitations
The framework supplies synchronization conditions for cellularly motivated nonlinear networks, with state-space and biochemical applications derived as corollaries.
Takeaways & Limitations
The state-space results assume locally Lipschitz dynamics, continuous outputs, global well-posedness, and arbitrary initial conditions subject to the stated reachability framework.
Abstract
from arXiv · showhide
This paper provides synchronization conditions for networks of nonlinear systems. The components of the network (referred to as "compartments'' in this paper) are made up of an identical interconnection of subsystems, each represented as an operator in an extended L2 space and referred to as a "species''. The compartments are, in turn, coupled through a diffusion-like term among the respective species. The synchronization conditions are provided by combining the input-output properties of the subsystems with information about the structure of network. The paper also explores results for state-space models, as well as biochemical applications. The work is motivated by cellular networks where signaling occurs both internally, through interactions of species, and externally, through intercellular signaling. The theory is illustrated providing synchronization conditions for networks of Goodwin oscillators.
1 Introduction
The paper studies synchronization in nonlinear-system networks by combining subsystem input-output properties with network structure. It targets cellularly motivated networks with internal species interactions and external intercompartment signaling, extending prior results beyond restrictive settings.
- The paper analyzes synchronization in networks of nonlinear systems using the input-output properties of their constituent subsystems.
- Each compartment contains interconnected species modeled as operators in extended L2, while identical species across compartments interact through diffusion-like coupling.
- Prior input-output studies addressed stability of individual compartments through passivity properties and dissipativity-matrix diagonal stability.
- The present approach requires only output-feedback passivity, permits noncyclic species interconnections, and uses minimal knowledge of physical laws for uncertain biological systems.
- The paper derives state-space corollaries, extends the framework to broader coupling structures, and illustrates it with Goodwin oscillator networks.
2 Preliminaries and problem statement
The model consists of identical compartments whose species are input-output operators, coupled internally through species interactions and externally through species-specific graph couplings. Synchrony is measured from deviations among compartment outputs, and the analysis uses relaxed cocoercivity, Laplacians, and dissipativity-matrix stability.
- Each of n identical compartments contains N species, with species k in compartment j represented by an operator Hk mapping inputs to outputs.
- Species coupling is encoded by the common interconnection matrix Σ, while nonnegative coefficients define compartmental coupling among identical species across compartments.
- Compartmental coupling uses output differences between compartments and is more general than true diffusion, which requires symmetric coupling coefficients.
- The framework characterizes connectivity through algebraic connectivity of directed coupling graphs and operator behavior through relaxed cocoercivity and its gain γ.
- The output disagreement norm ||∆Yk||T measures synchrony because it vanishes exactly when all outputs of species k are equal across compartments.
- The dissipativity matrix combines species interconnection information with cocoercivity gains, extending diagonal-stability analysis from compartment stability to synchronization.
3 Main results
The main theorem bounds output desynchronization from external-input desynchronization under relaxed-cocoercivity and diagonal-stability conditions. State-space corollaries further yield asymptotic output synchronization, and under detectability, state synchronization toward the isolated system’s limit set.
- Small external-input disagreement produces small output disagreement when the operators are relaxed cocoercive and the interconnection conditions hold.
- The theorem requires each Hk to be γk-relaxed cocoercive, λk + γk > 0, and the augmented dissipativity matrix to be diagonally stable.
- Positive algebraic connectivity augments species co-coercivity gains and can establish synchronization even when isolated compartments fail the earlier stability test.
- For state-space models, zero initial-condition input-output operators allow the theorem to apply, while arbitrary initial conditions require reachability and detectability assumptions.
- With no external inputs, outputs of each species converge pairwise across compartments; under the stated state-output condition, bounded network states synchronize.
- The synchronized solution converges to the limit set of the isolated system, where compartments are uncoupled.
4 Proof of the main result and corollary
The proof establishes an input-output inequality for disagreement variables and uses network coupling, diagonal stability, and reachability arguments to derive output and state synchronization.
- Proof of the main result: Pairwise relaxed cocoercivity yields nonnegative disagreement products, which are rewritten using the projection matrix and averaged outputs.The proof obtains ⟨˜Zk, ˜Yk⟩T = ⟨Zk,Yk⟩T − n⟨¯Zk,¯Yk⟩T ≥ 0.
- Proof of the main result: Diffusive coupling augments each cocoercivity coefficient from γk to ˜γk = γk + λk, while the coupling matrices are represented through Laplacians.The λk terms arise from the reduced Laplacian matrices and their algebraic connectivity.
- Proof of the corollary: State synchronization follows when state trajectories asymptotically agree, and bounded synchronized solutions converge to the limit set of the isolated system.The synchronized solution retains the isolated-system limit set because each Laplacian satisfies Lk1n = 0.
5 Discussion and extensions
The paper develops verifiable relaxed-cocoercivity conditions for ODE and memoryless operators, then extends synchronization analysis to state coupling, output coupling, and arbitrary initial conditions. Under cocoercivity, reachability, detectability, and graph conditions, compartment outputs synchronize and, in stronger cases, states and limit behavior converge.
- 5.1 Conditions for relaxed cocoercivity: Relaxed cocoercivity of input-output operators must generally be checked case by case, motivating verifiable conditions for ODE-based systems.The paper derives such conditions for particular ODE classes.
- 5.1 Conditions for relaxed cocoercivity: For a one-dimensional ODE, integrating the incremental energy inequality establishes relaxed cocoercivity of the associated input-output operator.The proof compares two trajectories, bounds the derivative of their squared difference, and integrates over a finite horizon.
- 5.1 Conditions for relaxed cocoercivity: Linear time-invariant specialization yields γ = a/b for the system ˙x = −ax + bu, y = x.This provides an explicit cocoercivity gain for a representative stable first-order system.
- 5.1 Conditions for relaxed cocoercivity: A monotone increasing Lipschitz static nonlinearity is shown to be ξ-relaxed cocoercive, and static nonlinearities can be analyzed directly through the defining inequality.The paper treats memoryless operators as a separate class of verifiable examples.
- 5.2 State coupling versus output coupling: The framework is motivated by biochemical networks, where compartmental coupling represents diffusion of reagent concentrations and may involve variables distinct from species outputs.The paper notes that coupling through nonlinear output functions can be unrealistic for some biochemical models.
- 5.2 State coupling versus output coupling: Theorem 2 handles output coupling when the dynamics operators and output nonlinearities are relaxed cocoercive, replacing γk with γk + ξkλk in the diagonal-stability condition.The condition also requires balanced Laplacians and diagonal stability of E˜γ.
- 5.2 State coupling versus output coupling: With zero-state reachability and the theorem’s assumptions, arbitrary-initial-condition networks synchronize in output; additional trajectory separation conditions yield state synchronization and convergence toward the isolated system’s limit set.The result applies to bounded network solutions and requires the stated implication from output convergence to state convergence.
- 5.2 State coupling versus output coupling: The same synchronization corollary can be interpreted for nonlinear observers by treating one compartment as a model and another as its state observer under unidirectional coupling.Diffusive coupling terms act as output injection in this interpretation.
6 Special structures and synchronization conditions
The paper specializes diagonal-stability-based synchronization conditions to cyclic and branched interconnection structures. For cyclic systems, the resulting condition generalizes an earlier first-species-only coupling result, while branched structures receive separate matrix conditions.
- General condition: Synchronization conditions are obtained by requiring E˜γ to be diagonally stable, linking compartmental algebraic connectivity with species interconnection structure.The inequalities also involve the relaxed cocoercivity gains of the species.
- Cyclic systems: For cyclic negative-feedback structures, the diagonal-stability test recovers a secant-type condition after incorporating the augmented gains ˜γk = γk + λk.The paper contrasts this with earlier synchronization work restricted to diffusion through one species.
- Cyclic systems: The cyclic result generalizes the earlier case in which compartmental coupling is limited to the first species, reducing to the expression reported in under that restriction.The paper explicitly notes the reciprocal convention for γk relative to.
- Cyclic systems: When every species is directly coupled across compartments with common weight q, the algebraic connectivity specializes to λk = q n for each compartment coupling.The positivity requirement then imposes q > −γk/n.
- Branched structures: For the first branched interconnection structure, diagonal stability of the associated dissipativity matrix is characterized through a condition supplied by Lemma 2 in.The paper then gives the corresponding specialization when coupling is limited to the first species.
- Branched structures: The second branched structure is analyzed by constructing its interconnection and dissipativity matrices and invoking a sufficient diagonal-stability condition.The displayed condition combines terms involving the augmented gains of several species.
7 Synchronization in networks of Goodwin oscillators
The Goodwin-oscillator example applies the input-output synchronization theory to cyclic biochemical compartments, relating diffusion topology, coupling strength, and cell number to synchronization.
- Goodwin oscillator model: Each Goodwin oscillator is modeled as a compartment containing four cyclically interconnected subsystems representing biochemical repression.The end product represses DNA-to-mRNA transcription, which controls enzyme production.
- Goodwin oscillator model: The model combines cyclic interconnection inputs with diffusion among identical species across compartments.The subsystem interconnections use inputs such as u1,j = −y3,j, u2,j = y1,j, and u3,j = y2,j; diffusion supplies the compartmental coupling.
- Synchronization condition: The synchronization condition is a secant-type inequality involving passivity coefficients γk and algebraic connectivities λk.For this model, (γ1 + λ1)(γ2 + λ2)(γ3 + λ3) > c, with c = 1/[γ4 sec(π/4)^4] ≅ 1.06.
- Synchronization condition: Under the synchronization condition and zero external inputs, species concentrations in different compartments synchronize; isolated compartments have unique equilibria.The isolated equilibria are asymptotically stable for p < 16, while p = 16 produces a Hopf bifurcation.
- Graph-topology effects: For complete coupling, increasing the number of cells can restore synchronization when a chosen diffusivity is insufficient, making cell number an order parameter.The simulations compare two oscillators, which do not synchronize, with 180 oscillators, which do.
- Graph-topology effects: Star, ring, and line topologies impose different synchronization requirements: stars require coupling above a threshold, whereas larger rings or lines can become more restrictive.For first- and second-species coupling, q > 0.074 for rings and q > 0.148 for lines yields an upper bound on cell number.
- Graph-topology effects: For a ring with q = 0.15, four oscillators synchronize but increasing the network to 45 cells does not produce synchronization.This illustrates the predicted restriction associated with increasing graph diameter.
- Unidirectional coupling: A unidirectional link coupling the first species of two Goodwin oscillators can be interpreted through a system-observer structure.Under the stated condition, the observer errors xk − x̂k converge to zero for k = 1, 2, 3.
8 Conclusion and future work
The paper combines subsystem input-output properties with network structure to study synchronization, derives state-space and biochemical corollaries, and identifies diffusion-model extensions as future work.
- The proposed framework combines input-output properties of nonlinear subsystems with network-structure information to analyze synchronization.
- State-space results and biochemical applications are derived as corollaries of the main synchronization result.
- Extending the work to diffusion models using partial differential operators is identified as ongoing future work.