1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.

/pdf/synchronization-of-chaotic-systems-5eqk79oyoj.pdf

Synchronization of chaotic systems.

Synchronization is of central importance in power distribution, telecommunication, neuronal and biological networks. Many networks are observed to produce patterns of synchronized clusters, but it has been difficult to predict these clusters or understand the conditions under which they form. Here we present a new framework and develop techniques for the analysis of network dynamics that shows the connection between network symmetries and cluster formation. The connection between symmetries and cluster synchronization is experimentally confirmed in the context of real networks with heterogeneities and noise using an electro-optic network. We experimentally observe and theoretically predict a surprising phenomenon in which some clusters lose synchrony without disturbing the others. Our analysis shows that such behaviour will occur in a wide variety of networks and node dynamics. The results could guide the design of new power grid systems or lead to new understanding of the dynamical behaviour of networks ranging from neural to social.

/pdf/cluster-synchronization-and-isolated-desynchronization-in-2eygmv966p.pdf

Cluster synchronization and isolated desynchronization in complex networks with symmetries

This paper investigates the dependence of synchronization transitions of bursting oscillations on the information transmission delay over scale-free neuronal networks with attractive and repulsive coupling. It is shown that for both types of coupling, the delay always plays a subtle role in either promoting or impairing synchronization. In particular, depending on the inherent oscillation period of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions are manifested as well-expressed minima in the measure for spatiotemporal synchrony. For attractive coupling, the minima appear at every integer multiple of the average oscillation period, while for the repulsive coupling, they appear at every odd multiple of the half of the average oscillation period. The obtained results are robust to the variations of the dynamics of individual neurons, the system size, and the neuronal firing type. Hence, they can be used to characterize attractively or repulsively coupled scale-free neuronal networks with delays.

/pdf/synchronous-bursts-on-scale-free-neuronal-networks-with-35x1qxoho7.pdf

Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling.

We investigate front propagation and synchronization transitions in dependence on the information transmission delay and coupling strength over scale-free neuronal networks with different average degrees and scaling exponents. As the underlying model of neuronal dynamics, we use the efficient Rulkov map with additive noise. We show that increasing the coupling strength enhances synchronization monotonously, whereas delay plays a more subtle role. In particular, we found that depending on the inherent oscillation frequency of individual neurons, regions of irregular and regular propagating excitatory fronts appear intermittently as the delay increases. These delay-induced synchronization transitions manifest as well-expressed minima in the measure for spatial synchrony, appearing at every multiple of the oscillation frequency. Larger coupling strengths or average degrees can broaden the region of regular propagating fronts by a given information transmission delay and further improve synchronization. These results are robust against variations in system size, intensity of additive noise, and the scaling exponent of the underlying scale-free topology. We argue that fine-tuned information transmission delays are vital for assuring optimally synchronized excitatory fronts on complex neuronal networks and, indeed, they should be seen as important as the coupling strength or the overall density of interneuronal connections. We finally discuss some biological implications of the presented results.

Synchronization transitions on scale-free neuronal networks due to finite information transmission delays.

Synchronous behavior in networks of coupled oscillators is a commonly observed phenomenon attracting a growing interest in physics, biology, communication, and other fields of science and technology. Besides global synchronization, one can also observe splitting of the full network into several clusters of mutually synchronized oscillators. In this paper, we study the conditions for such cluster partitioning into ensembles for the case of identical chaotic systems. We focus mainly on the existence and the stability of unique unconditional clusters whose rise does not depend on the origin of the other clusters. Also, conditional clusters in arrays of globally nonsymmetrically coupled identical chaotic oscillators are investigated. The design problem of organizing clusters into a given configuration is discussed.

Cluster synchronization in oscillatory networks.

We study collective phenomena in highly heterogeneous cardiac cell culture and its models. A cardiac culture is a mixture of passive (fibroblasts), oscillatory (pacemakers), and excitable (myocytes) cells. There is also heterogeneity within each type of cell as well. Results of in vitro experiments are modelled by Luo–Rudy and FitzHugh–Nagumo systems. For oscillatory and excitable media, we focus on the transitions from fully incoherent behavior to partially coherent behavior and then to global synchronization as the coupling strength is increased. These regimes are characterized qualitatively by spatiotemporal diagrams and quantitatively by profiles of dependence of individual frequencies on coupling. We find that synchronization clusters are determined by concentric and spiral waves. These waves arising due to the heterogeneity of medium push covered cells to oscillate in synchrony. We are also interested in the influence of passive and excitable elements on the oscillatory characteristics of low- and high-dimensional ensembles of cardiac cells. The mixture of initially silent excitable and passive cells shows the transitions to oscillatory behavior. In the media of oscillatory and passive or excitable cells, the effect of oscillation death is observed.

Synchronization phenomena in mixed media of passive, excitable, and oscillatory cells.

Spatiotemporal control of heart rate in a rabbit heart

We consider a three-domain model of cardiac tissue consisting of fibroblasts, myocytes, and extracellular space. We show in the one dimensional case that the fibroblasts with different resting potentials may alter restitution properties of tissue. On this basis we demonstrated that in two dimensional slice of cardiac tissue, a spiral wave break up can be caused purely by the influence of fibroblasts and, vice-versa, initially unstable spiral can be stabilized by fibroblasts depending on the value of their resting potential.

Fibroblasts alter spiral wave stability.

This paper presents the results of the development of a biomedical software system at the Nizhni Novgorod State University High Performance Computing Competence Center. We consider four main fields: plasma simulation, heart activity simulation, brain sensing simulation, molecular dynamics simulation. The software system is aimed at large-scale simulation on cluster systems with high efficiency and scalability. We demonstrate current results of the numerical simulation, analyze performance and propose the ways to improve efficiency.

V. S. Petrov

Papers

Cluster synchronization in oscillatory networks.

Synchronization phenomena in mixed media of passive, excitable, and oscillatory cells.

Spatiotemporal control of heart rate in a rabbit heart

Fibroblasts alter spiral wave stability.

High Performance Computing in Biomedical Applications