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1. Using noise to augment synchronization among oscillators

   https://doi.org/10.1038/s41598-021-83806-9  

   Vaidya, Jaykumar; Bashar, Mohammad Khairul; Shukla, Nikhil (December 2021, Scientific Reports)

   null (Ed.)

   Abstract Noise is expected to play an important role in the dynamics of analog systems such as coupled oscillators which have recently been explored as a hardware platform for application in computing. In this work, we experimentally investigate the effect of noise on the synchronization of relaxation oscillators and their computational properties. Specifically, in contrast to its typically expected adverse effect, we first demonstrate that a common white noise input induces frequency locking among uncoupled oscillators. Experiments show that the minimum noise voltage required to induce frequency locking increases linearly with the amplitude of the oscillator output whereas it decreases with increasing number of oscillators. Further, our work reveals that in a coupled system of oscillators—relevant to solving computational problems such as graph coloring, the injection of white noise helps reduce the minimum required capacitive coupling strength. With the injection of noise, the coupled system demonstrates frequency locking along with the desired phase-based computational properties at 5 × lower coupling strength than that required when no external noise is introduced. Consequently, this can reduce the footprint of the coupling element and the corresponding area-intensive coupling architecture. Our work shows that noise can be utilized as an effective knob to optimize the implementation of coupled oscillator-based computing platforms. 

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2. Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

   Menara, T.; Baggio, G.; Bassett, D; Pasqualetti, F (January 2019, IEEE transactions on control of network systems)

   In this paper we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the basis of several biological and technological processes; yet the underlying mechanisms to enable cluster synchronization of Kuramoto oscillators have remained elusive. In this paper we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intra-cluster coupling is sufficiently stronger than the inter-cluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intra-cluster dynamics is homogeneous. We illustrate and validate the effectiveness of our theoretical results via numerical studies. 

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3. Functional control of oscillator networks

   https://doi.org/10.1038/s41467-022-31733-2  

   Menara, Tommaso; Baggio, Giacomo; Bassett, Dani; Pasqualetti, Fabio (August 2022, Nature Communications)

   Abstract Oscillatory activity is ubiquitous in natural and engineered network systems. The interaction scheme underlying interdependent oscillatory components governs the emergence of network-wide patterns of synchrony that regulate and enable complex functions. Yet, understanding, and ultimately harnessing, the structure-function relationship in oscillator networks remains an outstanding challenge of modern science. Here, we address this challenge by presenting a principled method to prescribe exact and robust functional configurations from local network interactions through optimal tuning of the oscillators’ parameters. To quantify the behavioral synchrony between coupled oscillators, we introduce the notion offunctional pattern, which encodes the pairwise relationships between the oscillators’ phases. Our procedure is computationally efficient and provably correct, accounts for constrained interaction types, and allows to concurrently assign multiple desired functional patterns. Further, we derive algebraic and graph-theoretic conditions to guarantee the feasibility and stability of target functional patterns. These conditions provide an interpretable mapping between the structural constraints and their functional implications in oscillator networks. As a proof of concept, we apply the proposed method to replicate empirically recorded functional relationships from cortical oscillations in a human brain, and to redistribute the active power flow in different models of electrical grids. 

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4. Hypernetwork modeling and topology of high-order interactions for complex systems

   https://doi.org/10.1073/pnas.2412220121  

   Feng, Li; Gong, Huiying; Zhang, Shen; Liu, Xiang; Wang, Yu; Che, Jincan; Dong, Ang; Griffin, Christopher H; Gragnoli, Claudia; Wu, Jie; et al (October 2024, Proceedings of the National Academy of Sciences)

   Interactions among the underlying agents of a complex system are not only limited to dyads but can also occur in larger groups. Currently, no generic model has been developed to capture high-order interactions (HOI), which, along with pairwise interactions, portray a detailed landscape of complex systems. Here, we integrate evolutionary game theory and behavioral ecology into a unified statistical mechanics framework, allowing all agents (modeled as nodes) and their bidirectional, signed, and weighted interactions at various orders (modeled as links or hyperlinks) to be coded into hypernetworks. Such hypernetworks can distinguish between how pairwise interactions modulate a third agent (active HOI) and how the altered state of each agent in turn governs interactions between other agents (passive HOI). The simultaneous occurrence of active and passive HOI can drive complex systems to evolve at multiple time and space scales. We apply the model to reconstruct a hypernetwork of hexa-species microbial communities, and by dissecting the topological architecture of the hypernetwork using GLMY homology theory, we find distinct roles of pairwise interactions and HOI in shaping community behavior and dynamics. The statistical relevance of the hypernetwork model is validated using a series of in vitro mono-, co-, and tricultural experiments based on three bacterial species. 

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5. Oscillations and coupling in interconnections of two-dimensional brain networks

   E. Nozari, J. Cortés (July 2019, Proceedings of the American Control Conference)

   Oscillations in the brain are one of the most ubiquitous and robust patterns of activity and correlate with various cognitive phenomena. In this work, we study the existence and properties of oscillations in simple mean-field models of brain activity with bounded linear-threshold rate dynamics. First, we obtain exact conditions for the existence of limit cycles in two-dimensional excitatory-inhibitory networks (E-I pairs) and provide generalizations for networks with one inhibitory and multiple excitatory nodes. Building on these results, we study networks of multiple E-I pairs, provide exact conditions for the lack of stable equilibria, and numerically show that this is a tight proxy for the existence of oscillatory behavior. Finally, we study cross-frequency coupling between pairs of oscillators each consisting of an E-I pair. We find that while both phase-phase coupling (synchronization) and phase-amplitude coupling (PAC) monotonically increase with inter-oscillator connection strength, there exists a tradeoff in increasing frequency mismatch between the oscillators as it de-synchronizes them while enhancing their PAC. 

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