Juan G. Restrepo

Onset of synchronization in large networks of coupled oscillators

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.

Characterizing the dynamical importance of network nodes and links.

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.

Calsequestrin-Mediated Mechanism for Cellular Calcium Transient Alternans

Intracellular calcium transient alternans (CTA) has a recognized role in arrhythmogenesis, but its origin is not yet fully understood. Recent models of CTA are based on a steep relationship between calcium release from the sarcoplasmic reticulum (SR) and its calcium load before release. This mechanism alone, however, does not explain recent observations of CTA without diastolic SR calcium content alternations. In addition, nanoscopic imaging of calcium dynamics has revealed that the elementary calcium release units of the SR can become refractory independently of their local calcium content. Here we show using a new physiologically detailed mathematical model of calcium cycling that luminal gating of the calcium release channels (RyRs) mediated by the luminal buffer calsequestrin (CSQN) can cause CTA independently of the steepness of the release-load relationship. In this complementary mechanism, CTA is caused by a beat-to-beat alternation in the number of refractory RyR channels and can occur with or without diastolic SR calcium content alternans depending on pacing conditions and uptake dynamics. The model has unique features, in that it treats a realistic number of spatially distributed and diffusively coupled dyads, each one with a realistic number of RyR channels, and that luminal CSQN buffering and gating is incorporated based on experimental data that characterizes the effect of the conformational state of CSQN on its buffering properties. In addition to reproducing observed features of CTA, this multiscale model is able to describe recent experiments in which CSQN expression levels were genetically altered as well as to reproduce nanoscopic measurements of spark restitution properties. The ability to link microscopic properties of the calcium release units to whole cell behavior makes this model a powerful tool to investigate the arrhythmogenic role of abnormal calcium handling in many pathological settings.

Approximating the largest eigenvalue of network adjacency matrices

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.

Predicting criticality and dynamic range in complex networks: effects of topology.

The collective dynamics of a network of coupled excitable systems in response to an external stimulus depends on the topology of the connections in the network. Here we develop a general theoretical approach to study the effects of network topology on dynamic range, which quantifies the range of stimulus intensities resulting in distinguishable network responses. We find that the largest eigenvalue of the weighted network adjacency matrix governs the network dynamic range. When the largest eigenvalue is exactly one, the system is in a critical state and its dynamic range is maximized. Further, we examine higher order behavior of the steady state system, which predicts that networks with more homogeneous degree distributions should have higher dynamic range. Our analysis, confirmed by numerical simulations, generalizes previous studies in terms of the largest eigenvalue of the adjacency matrix.

Downlink Performance Analysis for a Generalized Shotgun Cellular System

In this paper, we analyze the signal-to-interference-plus-noise ratio (SINR) performance at a mobile station (MS) in a random cellular network. The cellular network is formed by base-stations (BSs) placed in a one, two or three dimensional space according to a possibly non-homogeneous Poisson point process, which is a generalization of the so-called shotgun cellular system. We develop a sequence of equivalence relations for the SCSs and use them to derive semi-analytical expressions for the coverage probability at the MS when the transmissions from each BS may be affected by random fading with arbitrary distributions as well as attenuation following arbitrary path-loss models. For homogeneous Poisson point processes in the interference-limited case with power-law path-loss model, we show that the SINR distribution is the same for all fading distributions and is not a function of the base station density. In addition, the influence of random transmission powers, power control, multiple channel reuse groups on the downlink performance are also discussed. The techniques developed for the analysis of SINR have applications beyond cellular networks and can be used in similar studies for cognitive radio networks, femtocell networks and other heterogeneous and multi-tier networks.

Synchronization in large directed networks of coupled phase oscillators.

We study the emergence of collective synchronization in large directed networks of heterogeneous oscillators by generalizing the classical Kuramoto model of globally coupled phase oscillators to more realistic networks. We extend recent theoretical approximations describing the transition to synchronization in large undirected networks of coupled phase oscillators to the case of directed networks. We also consider the case of networks with mixed positive-negative coupling strengths. We compare our theory with numerical simulations and find good agreement.

Spatiotemporal intracellular calcium dynamics during cardiac alternans

Cellular calcium transient alternans are beat-to-beat alternations in the peak cytosolic calcium concentration exhibited by cardiac cells during rapid electrical stimulation or under pathological conditions. Calcium transient alternans promote action potential duration alternans, which have been linked to the onset of life-threatening ventricular arrhythmias. Here we use a recently developed physiologically detailed mathematical model of ventricular myocytes to investigate both stochastic and deterministic aspects of intracellular calcium dynamics during alternans. The model combines a spatially distributed description of intracellular calcium cycling, where a large number of calcium release units are spatially distributed throughout the cell, with a full set of ionic membrane currents. The results demonstrate that ion channel stochasticity at the level of single calcium release units can influence the whole-cell alternans dynamics by causing phase reversals over many beats during fixed frequency pacing close to the alternans bifurcation. They also demonstrate the existence of a wide range of dynamical states. Depending on the sign and magnitude of calcium-voltage coupling, calcium alternans can be spatially synchronized or desynchronized, in or out of phase with action potential duration alternans, and the node separating out-of-phase regions of calcium alternans can be expelled from or trapped inside the cell. This range of states is found to be larger than previously anticipated by including a robust global attractor where calcium alternans can be spatially synchronized but out of phase with action potential duration alternans. The results are explained by a combined theoretical analysis of alternans stability and node motion using general iterative maps of the beat-to-beat dynamics and amplitude equations.

Downlink coverage analysis in a heterogeneous cellular network

In this paper, we consider the downlink signal-to-interference-plus-noise ratio (SINR) analysis in a heterogeneous cellular network with K tiers. Each tier is characterized by a base-station (BS) arrangement according to a homogeneous Poisson point process with certain BS density, transmission power, random shadow fading factors with arbitrary distribution, arbitrary path-loss exponent and a certain bias towards admitting the mobile-station (MS). The MS associates with the BS that has the maximum instantaneous biased received power under the open access cell association scheme. For such a general setting, we provide an analytical characterization of the coverage probability at the MS.

Hierarchical synchrony of phase oscillators in modular networks

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.