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The Hamiltonian mean field model of coupled inertial Hamiltonian rotors is a prototype for conservative dynamics in systems with long-range interactions. We consider the case where the interactions between the rotors are governed by a network described by a weighted adjacency matrix. By studying the linear stability of the incoherent state, we find that the transition to synchrony begins when the coupling constant K is inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters to study the effect of network heterogeneity on the synchronization of the rotors. When K is just beyond the transition to synchronization, we find that the degree of synchronization is highly dependent on the network's heterogeneity, but that for large K the degree of synchronization is robust to changes in the degree distribution. Our results are illustrated with numerical simulations on Erdös-Renyi networks and networks with power-law degree distributions.

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.

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.

We study the Hamiltonian mean field (HMF) model of coupled Hamiltonian rotors with a heterogeneous distribution of moments of inertia and coupling strengths. We show that when the parameters of the rotors are heterogeneous, finite-size fluctuations can greatly modify the coupling strength at which the incoherent state loses stability by inducing correlations between the momenta and parameters of the rotors. When the distribution of initial frequencies of the oscillators is sufficiently narrow, an analytical expression for the modification in critical coupling strength is obtained that confirms numerical simulations. We find that heterogeneity in the moments of inertia tends to stabilize the incoherent state, while heterogeneity in the coupling strengths tends to destabilize the incoherent state. Numerical simulations show that these effects disappear for a wide, bimodal frequency distribution.

We characterize the distributions of size and duration of avalanches propagating in complex networks. By an avalanche we mean the sequence of events initiated by the externally stimulated excitation of a network node, which may, with some probability, then stimulate subsequent excitations of the nodes to which it is connected, resulting in a cascade of excitations. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemiology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix that encodes the structure of the network. By using mean-field analyses, previous studies of avalanches in networks have not considered the effect of network structure on the distribution of size and duration of avalanches. Our results apply to individual networks (rather than network ensembles) and provide expressions for the distributions of size and duration of avalanches starting at particular nodes in the network. These findings might find application in the analysis of branching processes in networks, such as cascading power grid failures and critical brain dynamics. In particular, our results show that some experimental signatures of critical brain dynamics (i.e., power-law distributions of size and duration of neuronal avalanches) are robust to complex underlying network topologies.

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.

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.

The principal eigenvalue λ of a network's adjacency matrix often determines dynamics on the network (e.g., in synchronization and spreading processes) and some of its structural properties (e.g., robustness against failure or attack) and is therefore a good indicator for how "strongly" a network is connected. We study how λ is modified by the addition of a module, or community, which has broad applications, ranging from those involving a single modification (e.g., introduction of a drug into a biological process) to those involving repeated additions (e.g., power-grid and transit development). We describe how to optimally connect the module to the network to either maximize or minimize the shift in λ, noting several applications of directing dynamics on networks.

We study the synchronization of Kuramoto oscillators with all-to-all coupling in the presence of slow, noisy frequency adaptation. In this paper, we develop a model for oscillators, which adapt both their phases and frequencies. It is found that this model naturally reproduces some observed phenomena that are not qualitatively produced by the standard Kuramoto model, such as long waiting times before the synchronization of clapping audiences. By assuming a self-consistent steady state solution, we find three stability regimes for the coupling constant k , separated by critical points k{1} and k{2}: (i) for kk{2}, the incoherent state becomes unstable and only the synchronized state exists; and (iii) for k{1}coherent transition can be predicted by using Kramer's escape time formula and grows exponentially with the number of oscillators; (b) when the incoherent state is unstable (k>k{2}), the average waiting time grows logarithmically with the number of oscillators.

Spiral wave propagation in period-2 excitable media is accompanied by line defects, the locus of points with period-1 oscillations. Here we investigate spiral line defects in cardiac tissue where period-2 behavior has a known arrhythmogenic role. We find that the number of line defects, which is constrained to be an odd integer, is 3 for a freely rotating spiral, with and without meander, but 1 for a spiral anchored around a fixed heterogeneity. We interpret this finding analytically using a simple theory where spiral wave unstable modes with different numbers of line defects correspond to quantized solutions of a Helmholtz equation. Furthermore, the slow inward rotation of spiral line defects is described in different regimes.

The largest eigenvalue of the adjacency matrix of a network plays an important role in several network processes (e.g., synchronization of oscillators, percolation on directed networks, and linear stability of equilibria of network coupled systems). In this paper we develop approximations to the largest eigenvalue of adjacency matrices and discuss the relationships between these approximations. Numerical experiments on simulated networks are used to test our results.

We study the transition from incoherence to coherence in large networks of coupled phase oscillators. We present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength. We find that, under appropriate conditions, the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix. We show how, with an additional assumption, a mean-field approximation recently proposed is recovered from our results. We test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied. We find that our theory describes the transition well in situations in which the mean-field approximation fails. We study the finite-size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase.

We adapt a previous model and analysis method (the master stability function), extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators, to the case of oscillators that are similar but not exactly identical. We find that bubbling induced desynchronization bursts occur for some parameter values. These bursts have spatial patterns, which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor. We test the analysis of bursts by comparison with numerical experiments. In the case that no bursting occurs, we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators.