Patterns of synchronized clusters in adaptive networks

Cluster synchronization (CS), where groups of elements in a network exhibit synchronized dynamics, is a prevalent phenomenon in various complex systems, including robotic teams, biological systems, traffic flow, and neural networks. The brain, for instance, relies on synchronization of brain areas for cognitive functions, with functional assemblies displaying distinct synchronous oscillations. Adaptivity or plasticity, the time-varying nature of network connectivity, is another common feature of these systems, coevolving with node dynamics. While studies have explored the role of adaptation in promoting full synchronization, the impact of plasticity on CS in complex networks remains relatively under-explored, particularly given the often approximate nature of synchronization in real-world systems. This research directly addresses this gap. The authors aim to understand the effects of plasticity on CS in adaptive networks and the mechanisms driving the emergence of CS patterns. They explore several key questions: (i) Which CS patterns exist in networks with adaptation? (ii) Which of these patterns are attractive in the presence of adaptation? (iii) Can linear stability analysis be performed in a reduced space to manage the high dimensionality of the problem? Their approach employs the Master Stability Function (MSF) approach for dimensionality reduction, focusing on exact synchronization while acknowledging the robustness of their findings to a degree of heterogeneity.

Previous work has established that even non-evolving complex networks can support multiple coexisting CS patterns. Understanding how plasticity interacts with these patterns and influences their emergence is critical, particularly considering the potential for learning strategies to encode specific synchronization patterns in networks. Existing MSF approaches have been used to study the stability of fully synchronized regimes in plastic networks, but a general methodology for analyzing CS in multi-layer adaptive systems with delays has been lacking. This study addresses this methodological gap.

The authors investigate the dynamics of a general adaptive multi-layer network described by a set of delay differential equations. The network comprises N coupled nodes, each with an m-dimensional state vector. The dynamics include individual node behavior (Fᵢ), coupling between nodes (g), and L different types of connections (layers), each represented by an adjacency matrix A⁽ˡ⁾. The adaptive coupling is modeled by the matrix B(t), evolving according to an adaptation rule controlled by a nonlinear function H and a parameter ε representing the adaptation timescale. This framework encompasses a broad class of dynamical models, connection types, and plasticity rules. Identifying possible CS patterns is addressed by demonstrating that any equitable cluster partition of the adjacency matrix corresponds to a flow-invariant cluster synchronous solution. Stability analysis of CS solutions is a significant challenge due to the high dimensionality of the problem. The authors overcome this by utilizing a coarse-grained dynamical model (quotient network) with Q nodes, representing the equitable clusters. They linearize the system around a cluster synchronization state and analyze stability using the MSF approach. Dimensionality reduction is achieved through a coordinate transformation that separates perturbation modes, effectively decoupling the stability problem into lower-dimensional subproblems. This involves a simultaneous block diagonalization (SBD) to find a transformation matrix T, which separates longitudinal and transverse perturbations. The transverse perturbations, crucial for stability analysis, are then analyzed using the MSF approach via Transverse Lyapunov Exponents (TLEs). The dimensionality reduction is quantified, comparing the number of equations in the original system (N_orig) with the reduced system (N_red), showing significant reduction even for large N.

The methodology was validated through three case studies: 1. **Population coding: A plastic network of neural masses:** A fully connected network of Wilson-Cowan neural mass models with Hebbian plasticity was used to simulate visual memory encoding. Stimulating subgroups of nodes induced stable two-cluster solutions with β-γ oscillations, whose frequencies and phase lags were dependent on the coupling strength and number of stimulated nodes. Further experiments with three-cluster solutions demonstrated the capacity to encode multiple stimuli with different frequencies and phase lags. 2. **Propagation of tumor disease: A two-layer adaptive network model:** A two-layer multiplex network of phase oscillators modeled tumor development. The healthy state was fully synchronized, while pathological states exhibited multifrequency clusters. The authors investigated the influence of the proportion of initially pathological nodes (r) and an age parameter (β) on the network's health. They found that a stable healthy state was possible across a range of parameters, highlighting the multistability of the system. They demonstrated that a higher number of healthy clusters increases the probability of a pathological state. Robustness analysis confirmed the validity of the results even with network heterogeneity. 3. **Network of Lorenz systems with non-global coupling:** A small, dense Erdős-Rényi network of coupled Lorenz systems with Hebbian adaptation was analyzed. The MSF-based method successfully identified stable synchronization patterns, showing good agreement with direct network simulations but with significantly reduced computational cost. Across all three case studies, the MSF-based method facilitated efficient stability analysis of cluster synchronization solutions, demonstrating its effectiveness in handling the dimensionality challenges inherent in complex adaptive networks. The authors showed how the methodology allows identifying which CS patterns are selected by the adaptation rules and characterizing the stability of different CS solutions.

The results demonstrate the power of the MSF-based dimensionality reduction technique in analyzing CS in adaptive networks. This method extends previous work on cluster synchronization to networks with adaptive connections of various types, heterogeneous nodes, and delays. The findings have implications for neuroscience and machine learning, where CS plays a vital role in neural processes, and plasticity can support learning strategies to encode information. The case studies highlight the diverse applications of the methodology, showcasing its utility in modeling neural memory, tumor disease, and general adaptive network dynamics. The observed multistability in these systems emphasizes the complexity of synchronization scenarios in adaptive networks.

This paper presents a novel general methodology for analyzing cluster synchronization in multi-layer adaptive networks. The MSF-based approach significantly reduces the complexity of stability analysis, enabling the study of multi-stable CS patterns. The applications to neural mass models, tumor disease, and chaotic oscillators illustrate the broad applicability and effectiveness of the proposed method. Future research could explore the extension of this framework to more heterogeneous networks and the development of control strategies based on the identified multi-stable states.

The method is primarily focused on exact cluster synchronization. While the authors demonstrate robustness to some levels of heterogeneity, highly heterogeneous networks might require modifications or alternative approaches. The effectiveness of the dimensionality reduction is also influenced by network characteristics, potentially becoming less pronounced with highly diverse node dynamics.