1
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Allcroft TJ, Duong JT, Skardal PS, Kovarik ML. Microfluidic single-cell measurements of oxidative stress as a function of cell cycle position. Anal Bioanal Chem 2023; 415:6481-6490. [PMID: 37682313 DOI: 10.1007/s00216-023-04924-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/11/2023] [Revised: 07/24/2023] [Accepted: 08/28/2023] [Indexed: 09/09/2023]
Abstract
Single-cell measurements routinely demonstrate high levels of variation between cells, but fewer studies provide insight into the analytical and biological sources of this variation. This is particularly true of chemical cytometry, in which individual cells are lysed and their contents separated, compared to more established single-cell measurements of the genome and transcriptome. To characterize population-level variation and its sources, we analyzed oxidative stress levels in 1278 individual Dictyostelium discoideum cells as a function of exogenous stress level and cell cycle position. Cells were exposed to varying levels of oxidative stress via singlet oxygen generation using the photosensitizer Rose Bengal. Single-cell data reproduced the dose-response observed in ensemble measurements by CE-LIF, superimposed with high levels of heterogeneity. Through experiments and data analysis, we explored possible biological sources of this heterogeneity. No trend was observed between population variation and oxidative stress level, but cell cycle position was a major contributor to heterogeneity in oxidative stress. Cells synchronized to the same stage of cell division were less heterogeneous than unsynchronized cells (RSD of 37-51% vs 93%), and mitotic cells had higher levels of reactive oxygen species than interphase cells. While past research has proposed changes in cell size during the cell cycle as a source of biological noise, the measurements presented here use an internal standard to normalize for effects of cell volume, suggesting a more complex contribution of cell cycle to heterogeneity of oxidative stress.
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2
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Skardal PS, Restrepo JG. Detecting disturbances in network-coupled dynamical systems with machine learning. Chaos 2023; 33:103137. [PMID: 37903406 DOI: 10.1063/5.0169237] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2023] [Accepted: 10/05/2023] [Indexed: 11/01/2023]
Abstract
Identifying disturbances in network-coupled dynamical systems without knowledge of the disturbances or underlying dynamics is a problem with a wide range of applications. For example, one might want to know which nodes in the network are being disturbed and identify the type of disturbance. Here, we present a model-free method based on machine learning to identify such unknown disturbances based only on prior observations of the system when forced by a known training function. We find that this method is able to identify the locations and properties of many different types of unknown disturbances using a variety of known forcing functions. We illustrate our results with both linear and nonlinear disturbances using food web and neuronal activity models. Finally, we discuss how to scale our method to large networks.
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Affiliation(s)
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA
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3
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Adhikari S, Restrepo JG, Skardal PS. Synchronization of phase oscillators on complex hypergraphs. Chaos 2023; 33:033116. [PMID: 37003819 DOI: 10.1063/5.0116747] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2022] [Accepted: 02/17/2023] [Indexed: 06/19/2023]
Abstract
We study the effect of structured higher-order interactions on the collective behavior of coupled phase oscillators. By combining a hypergraph generative model with dimensionality reduction techniques, we obtain a reduced system of differential equations for the system's order parameters. We illustrate our framework with the example of a hypergraph with hyperedges of sizes 2 (links) and 3 (triangles). For this case, we obtain a set of two coupled nonlinear algebraic equations for the order parameters. For strong values of coupling via triangles, the system exhibits bistability and explosive synchronization transitions. We find conditions that lead to bistability in terms of hypergraph properties and validate our predictions with numerical simulations. Our results provide a general framework to study the synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features.
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Affiliation(s)
- Sabina Adhikari
- Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado 80309, USA
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4
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Skardal PS, Adhikari S, Restrepo JG. Multistability in coupled oscillator systems with higher-order interactions and community structure. Chaos 2023; 33:023140. [PMID: 36859233 DOI: 10.1063/5.0106906] [Citation(s) in RCA: 5] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2022] [Accepted: 01/30/2023] [Indexed: 06/18/2023]
Abstract
We study synchronization dynamics in populations of coupled phase oscillators with higher-order interactions and community structure. We find that the combination of these two properties gives rise to a number of states unsupported by either higher-order interactions or community structure alone, including synchronized states with communities organized into clusters in-phase, anti-phase, and a novel skew-phase, as well as an incoherent-synchronized state. Moreover, the system displays strong multistability with many of these states stable at the same time. We demonstrate our findings by deriving the low dimensional dynamics of the system and examining the system's bifurcations using stability analysis and perturbation theory.
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Affiliation(s)
| | - Sabina Adhikari
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA
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5
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Chamlagai PR, Taylor D, Skardal PS. Grass-roots optimization of coupled oscillator networks. Phys Rev E 2022; 106:034202. [PMID: 36266841 DOI: 10.1103/physreve.106.034202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2021] [Accepted: 08/16/2022] [Indexed: 06/16/2023]
Abstract
Despite the prevalence of biological and physical systems for which synchronization is critical, existing theory for optimizing synchrony depends on global information and does not sufficiently explore local mechanisms that enhance synchronization. Thus, there is a lack of understanding for the self-organized, collective processes that aim to optimize or repair synchronous systems, e.g., the dynamics of paracrine signaling within cardiac cells. Here we present "grass-roots" optimization of synchronization, which is a multiscale mechanism in which local optimizations of smaller subsystems cooperate to collectively optimize an entire system. Considering models of cardiac tissue and a power grid, we show that grass-roots-optimized systems are comparable to globally optimized systems, but they also have the added benefit of being robust to targeted attacks or subsystem islanding. Our findings motivate and support further investigation into the physics of local mechanisms that can support self-optimization for complex systems.
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Affiliation(s)
- Pranick R Chamlagai
- Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA
| | - Dane Taylor
- Department of Mathematics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
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6
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Skardal PS, Xu C. Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions. Chaos 2022; 32:053120. [PMID: 35649978 DOI: 10.1063/5.0086305] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
We study synchronization in large populations of coupled phase oscillators with time delays and higher-order interactions. With each of these effects individually giving rise to bistability between incoherence and synchronization via subcriticality at the onset of synchronization and the development of a saddle node, we find that their combination yields another mechanism behind bistability, where supercriticality at onset may be maintained; instead, the formation of two saddle nodes creates tiered synchronization, i.e., bistability between a weakly synchronized state and a strongly synchronized state. We demonstrate these findings by first deriving the low dimensional dynamics of the system and examining the system bifurcations using a stability and steady-state analysis.
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Affiliation(s)
| | - Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
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7
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Ziegler C, Skardal PS, Dutta H, Taylor D. Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes. Chaos 2022; 32:023128. [PMID: 35232052 DOI: 10.1063/5.0080370] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Accepted: 01/31/2022] [Indexed: 06/14/2023]
Abstract
Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional "simplices") and how they are influenced through higher-order interactions. A prime example lies in neuroscience where groups of neurons (not individual ones) may provide building blocks for neurocomputation. Here, we study consensus dynamics on edges in simplicial complexes using a type of Laplacian matrix called a Hodge Laplacian, which we generalize to allow higher- and lower-order interactions to have different strengths. Using techniques from algebraic topology, we study how collective dynamics converge to a low-dimensional subspace that corresponds to the homology space of the simplicial complex. We use the Hodge decomposition to show that higher- and lower-order interactions can be optimally balanced to maximally accelerate convergence and that this optimum coincides with a balancing of dynamics on the curl and gradient subspaces. We additionally explore the effects of network topology, finding that consensus over edges is accelerated when two-simplices are well dispersed, as opposed to clustered together.
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Affiliation(s)
- Cameron Ziegler
- Department of Mathematics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
| | | | - Haimonti Dutta
- Management Science and Systems Department, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
| | - Dane Taylor
- Department of Mathematics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
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8
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Arola-Fernández L, Skardal PS, Arenas A. Geometric unfolding of synchronization dynamics on networks. Chaos 2021; 31:061105. [PMID: 34241326 DOI: 10.1063/5.0053837] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/12/2021] [Accepted: 05/17/2021] [Indexed: 06/13/2023]
Abstract
We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.
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Affiliation(s)
- Lluís Arola-Fernández
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | | | - Alex Arenas
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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9
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Xu C, Wang X, Skardal PS. Universal scaling and phase transitions of coupled phase oscillator populations. Phys Rev E 2020; 102:042310. [PMID: 33212588 DOI: 10.1103/physreve.102.042310] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2020] [Accepted: 10/12/2020] [Indexed: 11/07/2022]
Abstract
The Kuramoto model, which serves as a paradigm for investigating synchronization phenomena of oscillatory systems, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here we generalize a number of classical results by presenting a general framework for analytically capturing the critical scaling properties of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on the coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determine the scaling properties of order parameter above criticality.
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Affiliation(s)
- Can Xu
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - Xuebin Wang
- Institute of Systems Science and College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
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10
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Taylor D, Skardal PS, Sun J. Introduction to Focus Issue: Symmetry and optimization in the synchronization and collective behavior of complex systems. Chaos 2020; 30:060401. [PMID: 32611075 DOI: 10.1063/5.0016491] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2020] [Accepted: 06/04/2020] [Indexed: 06/11/2023]
Abstract
Synchronization phenomena and collective behavior are commonplace in complex systems with applications ranging from biological processes such as coordinated neuron firings and cell cycles to the stability of alternating current power grids. A fundamental pursuit is the study of how various types of symmetry-e.g., as manifest in network structure or coupling dynamics-impact a system's collective behavior. Understanding the intricate relations between structural and dynamical symmetry/asymmetry also provides new paths to develop strategies that enhance or inhibit synchronization. Previous research has revealed symmetry as a key factor in identifying optimization mechanisms, but the particular ways that symmetry/asymmetry influence collective behavior can generally depend on the type of dynamics, networks, and form of synchronization (e.g., phase synchronization, group synchronization, and chimera states). Other factors, such as time delay, noise, time-varying structure, multilayer connections, basin stability, and transient dynamics, also play important roles, and many of these remain underexplored. This Focus Issue brings together a survey of theoretical and applied research articles that push forward this important line of questioning.
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Affiliation(s)
- D Taylor
- Department of Mathematics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA
| | - P S Skardal
- Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA
| | - J Sun
- Clarkson Center for Complex Systems Science, Potsdam, New York 13699, USA
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11
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Skardal PS. Quasiperiodic dynamics and a Neimark-Sacker bifurcation in nonlinear random walks on complex networks. Phys Rev E 2020; 101:012307. [PMID: 32069530 DOI: 10.1103/physreve.101.012307] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2019] [Indexed: 06/10/2023]
Abstract
We study the dynamics of nonlinear random walks on complex networks. In particular, we investigate the role and effect of directed network topologies on long-term dynamics. While a period-doubling bifurcation to alternating patterns occurs at a critical bias parameter value, we find that some directed structures give rise to a different kind of bifurcation that gives rise to quasiperiodic dynamics. This does not occur for all directed network structure, but only when the network structure is sufficiently directed. We find that the onset of quasiperiodic dynamics is the result of a Neimark-Sacker bifurcation, where a pair of complex-conjugate eigenvalues of the system Jacobian pass through the unit circle, destabilizing the stationary distribution with high-dimensional rotations. We investigate the nature of these bifurcations, study the onset of quasiperiodic dynamics as network structure is tuned to be more directed, and present an analytically tractable case of a four-neighbor ring.
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12
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Skardal PS, Arenas A. Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes. Phys Rev Lett 2019; 122:248301. [PMID: 31322386 DOI: 10.1103/physrevlett.122.248301] [Citation(s) in RCA: 74] [Impact Index Per Article: 14.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/28/2019] [Revised: 05/28/2019] [Indexed: 05/20/2023]
Abstract
Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.
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Affiliation(s)
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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13
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Abstract
The Kuramoto model is a paradigmatic tool for studying the dynamics of collective behavior in large ensembles of coupled dynamical systems. Over the past decade a great deal of progress has been made in analytical descriptions of the macroscopic dynamics of the Kuramoto model, facilitated by the discovery of Ott and Antonsen's dimensionality reduction method. However, the vast majority of these works relies on a critical assumption where the oscillators' natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which allows for a convenient closure of the evolution equations from the dimensionality reduction. In this paper we investigate the low-dimensional dynamics that emerge from a broader family of natural frequency distributions, in particular, a family of rational distribution functions. We show that, as the polynomials that characterize the frequency distribution increase in order, the low-dimensional evolution equations become more complicated, but nonetheless the system dynamics remain simple, displaying a transition from incoherence to partial synchronization at a critical coupling strength. Using the low-dimensional equations we analytically calculate the critical coupling strength corresponding to the onset of synchronization and investigate the scaling properties of the order parameter near the onset of synchronization. These results agree with calculations from Kuramoto's original self-consistency framework, but we emphasize that the low-dimensional equations approach used here allows for a true stability analysis categorizing the bifurcations.
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14
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Skardal PS. Diffusion dynamics and synchronizability of hierarchical products of networks. Phys Rev E 2017; 96:042302. [PMID: 29347571 DOI: 10.1103/physreve.96.042302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/10/2017] [Indexed: 06/07/2023]
Abstract
The hierarchical product of networks represents a natural tool for building large networks out of two smaller subnetworks: a primary subnetwork and a secondary subnetwork. Here we study the dynamics of diffusion and synchronization processes on hierarchical products. We apply techniques previously used for approximating the eigenvalues of the adjacency matrix to the Laplacian matrix, allowing us to quantify the effects that the primary and secondary subnetworks have on diffusion and synchronization in terms of a coupling parameter that weighs the secondary subnetwork relative to the primary subnetwork. Diffusion processes are separated into two regimes: for small coupling the diffusion rate is determined by the structure of the secondary network, scaling with the coupling parameter, while for large coupling it is determined by the primary network and saturates. Synchronization, on the other hand, is separated into three regimes, for both small and large coupling hierarchical products have poor synchronization properties, but is optimized at an intermediate value. Moreover, the critical coupling value that optimizes synchronization is shaped by the relative connectivities of the primary and secondary subnetworks, compensating for significant differences between the two subnetworks.
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15
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Skardal PS, Restrepo JG, Ott E. Uncovering low dimensional macroscopic chaotic dynamics of large finite size complex systems. Chaos 2017; 27:083121. [PMID: 28863484 DOI: 10.1063/1.4986957] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/20/2023]
Abstract
In the last decade, it has been shown that a large class of phase oscillator models admit low dimensional descriptions for the macroscopic system dynamics in the limit of an infinite number N of oscillators. The question of whether the macroscopic dynamics of other similar systems also have a low dimensional description in the infinite N limit has, however, remained elusive. In this paper, we show how techniques originally designed to analyze noisy experimental chaotic time series can be used to identify effective low dimensional macroscopic descriptions from simulations with a finite number of elements. We illustrate and verify the effectiveness of our approach by applying it to the dynamics of an ensemble of globally coupled Landau-Stuart oscillators for which we demonstrate low dimensional macroscopic chaotic behavior with an effective 4-dimensional description. By using this description, we show that one can calculate dynamical invariants such as Lyapunov exponents and attractor dimensions. One could also use the reconstruction to generate short-term predictions of the macroscopic dynamics.
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Affiliation(s)
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA
| | - Edward Ott
- Department of Physics and Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland 20742, USA
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16
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Nicosia V, Skardal PS, Arenas A, Latora V. Collective Phenomena Emerging from the Interactions between Dynamical Processes in Multiplex Networks. Phys Rev Lett 2017; 118:138302. [PMID: 28409987 DOI: 10.1103/physrevlett.118.138302] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2016] [Indexed: 05/28/2023]
Abstract
We introduce a framework to intertwine dynamical processes of different nature, each with its own distinct network topology, using a multilayer network approach. As an example of collective phenomena emerging from the interactions of multiple dynamical processes, we study a model where neural dynamics and nutrient transport are bidirectionally coupled in such a way that the allocation of the transport process at one layer depends on the degree of synchronization at the other layer, and vice versa. We show numerically, and we prove analytically, that the multilayer coupling induces a spontaneous explosive synchronization and a heterogeneous distribution of allocations, otherwise not present in the two systems considered separately. Our framework can find application to other cases where two or more dynamical processes such as synchronization, opinion formation, information diffusion, or disease spreading, are interacting with each other.
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Affiliation(s)
- Vincenzo Nicosia
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
| | | | - Alex Arenas
- Department d'Enginyeria Informática i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Vito Latora
- School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom
- Dipartimento di Fisica ed Astronomia, Università di Catania and INFN, I-95123 Catania, Italy
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Abstract
Today's colleges and universities consist of highly complex structures that dictate interactions between the administration, faculty, and student body. These structures can play a role in dictating the efficiency of policy enacted by the administration and determine the effect that curriculum changes in one department have on other departments. Despite the fact that the features of these complex structures have a strong impact on the institutions, they remain by-and-large unknown in many cases. In this paper we study the academic structure of our home institution of Trinity College in Hartford, CT using the major and minor patterns between graduating students to build a temporal multiplex network describing the interactions between different departments. Using recent network science techniques developed for such temporal networks we identify the evolving community structures that organize departments' interactions, as well as quantify the interdisciplinary centrality of each department. We implement this framework for Trinity College, finding practical insights and applications, but also present it as a general framework for colleges and universities to better understand their own structural makeup in order to better inform academic and administrative policy.
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Affiliation(s)
- Shufan Wang
- Department of Mathematics, Trinity College, 300 Summit St., 06106, West Hartford, Connecticut, USA
| | - Mariam Avagyan
- Department of Mathematics, Trinity College, 300 Summit St., 06106, West Hartford, Connecticut, USA
| | - Per Sebastian Skardal
- Department of Mathematics, Trinity College, 300 Summit St., 06106, West Hartford, Connecticut, USA
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18
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Skardal PS, Sevilla-Escoboza R, Vera-Ávila VP, Buldú JM. Optimal phase synchronization in networks of phase-coherent chaotic oscillators. Chaos 2017; 27:013111. [PMID: 28147498 DOI: 10.1063/1.4974029] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We investigate the existence of an optimal interplay between the natural frequencies of a group of chaotic oscillators and the topological properties of the network they are embedded in. We identify the conditions for achieving phase synchronization in the most effective way, i.e., with the lowest possible coupling strength. Specifically, we show by means of numerical and experimental results that it is possible to define a synchrony alignment function J(ω,L) linking the natural frequencies ωi of a set of non-identical phase-coherent chaotic oscillators with the topology of the Laplacian matrix L, the latter accounting for the specific organization of the network of interactions between oscillators. We use the classical Rössler system to show that the synchrony alignment function obtained for phase oscillators can be extended to phase-coherent chaotic systems. Finally, we carry out a series of experiments with nonlinear electronic circuits to show the robustness of the theoretical predictions despite the intrinsic noise and parameter mismatch of the electronic components.
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Affiliation(s)
- P S Skardal
- Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA
| | - R Sevilla-Escoboza
- Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de Leon, Paseos de la Montaña, Lagos de Moreno, Jalisco 47460, Mexico
| | - V P Vera-Ávila
- Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de Leon, Paseos de la Montaña, Lagos de Moreno, Jalisco 47460, Mexico
| | - J M Buldú
- Laboratory of Biological Networks, Center for Biomedical Technology, UPM, Pozuelo de Alarcón, 28223 Madrid, Spain
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19
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Skardal PS, Wash K. Spectral properties of the hierarchical product of graphs. Phys Rev E 2016; 94:052311. [PMID: 27967095 DOI: 10.1103/physreve.94.052311] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/13/2016] [Indexed: 06/06/2023]
Abstract
The hierarchical product of two graphs represents a natural way to build a larger graph out of two smaller graphs with less regular and therefore more heterogeneous structure than the Cartesian product. Here we study the eigenvalue spectrum of the adjacency matrix of the hierarchical product of two graphs. Introducing a coupling parameter describing the relative contribution of each of the two smaller graphs, we perform an asymptotic analysis for the full spectrum of eigenvalues of the adjacency matrix of the hierarchical product. Specifically, we derive the exact limit points for each eigenvalue in the limits of small and large coupling, as well as the leading-order relaxation to these values in terms of the eigenvalues and eigenvectors of the two smaller graphs. Given its central roll in the structural and dynamical properties of networks, we study in detail the Perron-Frobenius, or largest, eigenvalue. Finally, as an example application we use our theory to predict the epidemic threshold of the susceptible-infected-susceptible model on a hierarchical product of two graphs.
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Affiliation(s)
| | - Kirsti Wash
- Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA
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20
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Abstract
We study optimal synchronization of networks of coupled phase oscillators. We extend previous theory for optimizing the synchronization properties of undirected networks to the important case of directed networks. We derive a generalized synchrony alignment function that encodes the interplay between the network structure and the oscillators' natural frequencies and serves as an objective measure for the network's degree of synchronization. Using the generalized synchrony alignment function, we show that a network's synchronization properties can be systematically optimized. This framework also allows us to study the properties of synchrony-optimized networks, and in particular, investigate the role of directed network properties such as nodal in- and out-degrees. For instance, we find that in optimally rewired networks, the heterogeneity of the in-degree distribution roughly matches the heterogeneity of the natural frequency distribution, but no such relationship emerges for out-degrees. We also observe that a network's synchronization properties are promoted by a strong correlation between the nodal in-degrees and the natural frequencies of oscillators, whereas the relationship between the nodal out-degrees and the natural frequencies has comparatively little effect. This result is supported by our theory, which indicates that synchronization is promoted by a strong alignment of the natural frequencies with the left singular vectors corresponding to the largest singular values of the Laplacian matrix.
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Affiliation(s)
| | - Dane Taylor
- Department of Mathematics, Carolina Center for Interdisciplinary Applied Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, New York 13699, USA
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Abstract
The control of network-coupled nonlinear dynamical systems is an active area of research in the nonlinear science community. Coupled oscillator networks represent a particularly important family of nonlinear systems, with applications ranging from the power grid to cardiac excitation. Here, we study the control of network-coupled limit cycle oscillators, extending the previous work that focused on phase oscillators. Based on stabilizing a target fixed point, our method aims to attain complete frequency synchronization, i.e., consensus, by applying control to as few oscillators as possible. We develop two types of controls. The first type directs oscillators towards larger amplitudes, while the second does not. We present numerical examples of both control types and comment on the potential failures of the method.
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Affiliation(s)
| | - Alex Arenas
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Tarragona, Spain
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Abstract
We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, CT 06106, USA; Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Abstract
We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, CT 06106, USA
- Departament d’Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
| | - Alex Arenas
- Departament d’Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Skardal PS, Taylor D, Sun J, Arenas A. Collective frequency variation in network synchronization and reverse PageRank. Phys Rev E 2016; 93:042314. [PMID: 27176319 PMCID: PMC5131881 DOI: 10.1103/physreve.93.042314] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2015] [Indexed: 05/16/2023]
Abstract
A wide range of natural and engineered phenomena rely on large networks of interacting units to reach a dynamical consensus state where the system collectively operates. Here we study the dynamics of self-organizing systems and show that for generic directed networks the collective frequency of the ensemble is not the same as the mean of the individuals' natural frequencies. Specifically, we show that the collective frequency equals a weighted average of the natural frequencies, where the weights are given by an outflow centrality measure that is equivalent to a reverse PageRank centrality. Our findings uncover an intricate dependence of the collective frequency on both the structural directedness and dynamical heterogeneity of the network, and also reveal an unexplored connection between synchronization and PageRank, which opens the possibility of applying PageRank optimization to synchronization. Finally, we demonstrate the presence of collective frequency variation in real-world networks by considering the UK and Scandinavian power grids.
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Affiliation(s)
| | - Dane Taylor
- Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
- Department of Physics, Clarkson University, Potsdam, NY 13699, USA
| | - Alex Arenas
- Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Taylor D, Skardal PS, Sun J. SYNCHRONIZATION OF HETEROGENEOUS OSCILLATORS UNDER NETWORK MODIFICATIONS: PERTURBATION AND OPTIMIZATION OF THE SYNCHRONY ALIGNMENT FUNCTION. SIAM J Appl Math 2016; 76:1984-2008. [PMID: 27872501 PMCID: PMC5115605 DOI: 10.1137/16m1075181] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
Abstract
Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications-for which proper functionality depends sensitively on the extent of synchronization-there remains a lack of understanding for how systems can best evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.
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Affiliation(s)
- Dane Taylor
- Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599, USA; and Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC, 27709, USA
| | | | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, NY, 13699, USA; Department of Physics, Potsdam, NY, 13699, USA; Department of Computer Science, Potsdam, NY, 13699, USA
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Skardal PS, Arenas A. Control of coupled oscillator networks with application to microgrid technologies. Sci Adv 2015; 1:e1500339. [PMID: 26601231 PMCID: PMC4643801 DOI: 10.1126/sciadv.1500339] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/17/2015] [Accepted: 07/20/2015] [Indexed: 05/15/2023]
Abstract
The control of complex systems and network-coupled dynamical systems is a topic of vital theoretical importance in mathematics and physics with a wide range of applications in engineering and various other sciences. Motivated by recent research into smart grid technologies, we study the control of synchronization and consider the important case of networks of coupled phase oscillators with nonlinear interactions-a paradigmatic example that has guided our understanding of self-organization for decades. We develop a method for control based on identifying and stabilizing problematic oscillators, resulting in a stable spectrum of eigenvalues, and in turn a linearly stable synchronized state. The amount of control, that is, number of oscillators, required to stabilize the network is primarily dictated by the coupling strength, dynamical heterogeneity, and mean degree of the network, and depends little on the structural heterogeneity of the network itself.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, CT 06106, USA
- Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
- Corresponding author. E-mail:
| | - Alex Arenas
- Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Skardal PS, Restrepo JG, Ott E. Frequency assortativity can induce chaos in oscillator networks. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 91:060902. [PMID: 26172652 DOI: 10.1103/physreve.91.060902] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2015] [Indexed: 05/16/2023]
Abstract
We investigate the effect of preferentially connecting oscillators with similar frequency to each other in networks of coupled phase oscillators (i.e., frequency assortativity). Using the network Kuramoto model as an example, we find that frequency assortativity can induce chaos in the macroscopic dynamics. By applying a mean-field approximation in combination with the dimension reduction method of Ott and Antonsen, we show that the dynamics can be described by a low dimensional system of equations. We use the reduced system to characterize the macroscopic chaos using Lyapunov exponents, bifurcation diagrams, and time-delay embeddings. Finally, we show that the emergence of chaos stems from the formation of multiple groups of synchronized oscillators, i.e., meta-oscillators.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Mathematics, Trinity College, Hartford, Connecticut 06106, USA
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA
| | - Edward Ott
- Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
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Skardal PS, Taylor D, Sun J, Arenas A. Erosion of synchronization in networks of coupled oscillators. Phys Rev E Stat Nonlin Soft Matter Phys 2015; 91:010802. [PMID: 25679557 DOI: 10.1103/physreve.91.010802] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2014] [Indexed: 05/16/2023]
Abstract
We report erosion of synchronization in networks of coupled phase oscillators, a phenomenon where perfect phase synchronization is unattainable in steady state, even in the limit of infinite coupling. An analysis reveals that the total erosion is separable into the product of terms characterizing coupling frustration and structural heterogeneity, both of which amplify erosion. The latter, however, can differ significantly from degree heterogeneity. Finally, we show that erosion is marked by the reorganization of oscillators according to their node degrees rather than their natural frequencies.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Dane Taylor
- Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA and Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, New York, 13699, USA
| | - Alex Arenas
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Skardal PS, Restrepo JG. Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans. Chaos 2014; 24:043126. [PMID: 25554046 DOI: 10.1063/1.4901728] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper, we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes-or phase reversals-low-periodic dynamics prevail, while away from the nodes, the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Juan G Restrepo
- Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309, USA
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30
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Abstract
We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of a synchrony alignment function that encodes the interplay between network structure and oscillators' frequencies and that can be readily optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching between the heterogeneity of frequencies and network structure.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informatica i Matemátiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain and Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA
| | - Dane Taylor
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA and Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina 27709, USA and Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599, USA
| | - Jie Sun
- Department of Mathematics, Clarkson University, Potsdam, New York 13699, USA
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Abstract
We study explosive synchronization, a phenomenon characterized by first-order phase transitions between incoherent and synchronized states in networks of coupled oscillators. While explosive synchronization has been the subject of many recent studies, in each case strong conditions on the heterogeneity of the network, its link weights, or its initial construction are imposed to engineer a first-order phase transition. This raises the question of how robust explosive synchronization is in view of more realistic structural and dynamical properties. Here we show that explosive synchronization can be induced in mildly heterogeneous networks by the addition of quenched disorder to the oscillators' frequencies, demonstrating that it is not only robust to, but moreover promoted by, this natural mechanism. We support these findings with numerical and analytical results, presenting simulations of a real neural network as well as a self-consistency theory used to study synthetic networks.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
| | - Alex Arenas
- Departament d'Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
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Skardal PS, Karma A, Restrepo JG. Spatiotemporal dynamics of calcium-driven cardiac alternans. Phys Rev E Stat Nonlin Soft Matter Phys 2014; 89:052707. [PMID: 25353829 PMCID: PMC4404323 DOI: 10.1103/physreve.89.052707] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2014] [Indexed: 06/04/2023]
Abstract
We investigate the dynamics of spatially discordant alternans (SDA) driven by an instability of intracellular calcium cycling using both amplitude equations [P. S. Skardal, A. Karma, and J. G. Restrepo, Phys. Rev. Lett. 108, 108103 (2012)] and ionic model simulations. We focus on the common case where the bidirectional coupling of intracellular calcium concentration and membrane voltage dynamics produces calcium and voltage alternans that are temporally in phase. We find that, close to the alternans bifurcation, SDA is manifested as a smooth wavy modulation of the amplitudes of both repolarization and calcium transient (CaT) alternans, similarly to the well-studied case of voltage-driven alternans. In contrast, further away from the bifurcation, the amplitude of CaT alternans jumps discontinuously at the nodes separating out-of-phase regions, while the amplitude of repolarization alternans remains smooth. We identify universal dynamical features of SDA pattern formation and evolution in the presence of those jumps. We show that node motion of discontinuous SDA patterns is strongly hysteretic even in homogeneous tissue due to the novel phenomenon of "unidirectional pinning": node movement can only be induced towards, but not away from, the pacing site in response to a change of pacing rate or physiological parameter. In addition, we show that the wavelength of discontinuous SDA patterns scales linearly with the conduction velocity restitution length scale, in contrast to the wavelength of smooth patterns that scales sublinearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations due to the property that unidirectional node motion collapses multiple jumps accumulating in nodal regions into a single jump. Amplitude equation predictions are in good overall agreement with ionic model simulations. Finally, we briefly discuss physiological implications of our findings. In particular, we suggest that due to the tendency of conduction blocks to form near nodes, the presence of unidirectional pinning makes calcium-driven alternans potentially more arrhythmogenic than voltage-driven alternans.
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Affiliation(s)
- Per Sebastian Skardal
- Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, 43007 Tarragona, Spain
- Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA
| | - Alain Karma
- Physics Department and Center for Interdisciplinary Research on Complex Systems, Northeastern University, Boston, Massachusetts 02115, USA
| | - Juan G. Restrepo
- Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA
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Skardal PS, Karma A, Restrepo JG. Unidirectional pinning and hysteresis of spatially discordant alternans in cardiac tissue. Phys Rev Lett 2012; 108:108103. [PMID: 22463458 DOI: 10.1103/physrevlett.108.108103] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2011] [Indexed: 05/31/2023]
Abstract
Spatially discordant alternans is a widely observed pattern of voltage and calcium signals in cardiac tissue that can precipitate lethal cardiac arrhythmia. Using spatially coupled iterative maps of the beat-to-beat dynamics, we explore this pattern's dynamics in the regime of a calcium-dominated period-doubling instability at the single-cell level. We find a novel nonlinear bifurcation associated with the formation of a discontinuous jump in the amplitude of calcium alternans at nodes separating discordant regions. We show that this jump unidirectionally pins nodes by preventing their motion away from the pacing site following a pacing rate decrease but permitting motion towards this site following a rate increase. This unidirectional pinning leads to strongly history-dependent node motion that is strongly arrhythmogenic.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA.
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Skardal PS, Restrepo JG. Hierarchical synchrony of phase oscillators in modular networks. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:016208. [PMID: 22400644 DOI: 10.1103/physreve.85.016208] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2011] [Indexed: 05/18/2023]
Abstract
We study synchronization of sinusoidally coupled phase oscillators on networks with modular structure and a large number of oscillators in each community. Of particular interest is the hierarchy of local and global synchrony, i.e., synchrony within and between communities, respectively. Using the recent ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we find that the degree of local synchrony can be determined from a set of coupled low-dimensional equations. If the number of communities in the network is large, a low-dimensional description of global synchrony can be also found. Using these results, we study bifurcations between different types of synchrony. We find that, depending on the relative strength of local and global coupling, the transition to synchrony in the network can be mediated by local or global effects.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309, USA.
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Skardal PS, Ott E, Restrepo JG. Cluster synchrony in systems of coupled phase oscillators with higher-order coupling. Phys Rev E Stat Nonlin Soft Matter Phys 2011; 84:036208. [PMID: 22060476 DOI: 10.1103/physreve.84.036208] [Citation(s) in RCA: 41] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2011] [Indexed: 05/31/2023]
Abstract
We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen [Chaos 18, 037113 (2008)] and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold. Shaped by this manifold, there is an infinite family of steady-state distributions of oscillators, resulting in a high degree of multistability in the cluster asymmetry. We also show how through external forcing the degree of asymmetry can be controlled, and suggest that systems displaying cluster synchrony can be used to encode and store data.
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Affiliation(s)
- Per Sebastian Skardal
- Department of Applied Mathematics, University of Colorado at Boulder, Colorado 80309, USA.
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