Search

In traditional models of opinion dynamics, each agent in a network has an opinion and changes in opinions arise from pairwise (i.e., dyadic) interactions between agents. However, in many situations, groups of individuals can possess a collective opinion that may differ from the opinions of the individuals. In this paper, we study the effects of group opinions on opinion dynamics. We formulate a hypergraph model in which both individual agents and groups of 3 agents have opinions, and we examine how opinions evolve through both dyadic interactions and group memberships. In some parameter regimes, we find that the presence of group opinions can lead to oscillatory and excitable opinion dynamics. In the oscillatory regime, the mean opinion of the agents in a network has self-sustained oscillations. In the excitable regime, finite-size effects create large but short-lived opinion swings (as in social fads). We develop a mean-field approximation of our model and obtain good agreement with direct numerical simulations. We also show, both numerically and via our mean-field description, that oscillatory dynamics occur only when the number of dyadic and polyadic interactions per agent are not completely correlated. Our results illustrate how polyadic structures, such as groups of agents, can have important effects on collective opinion dynamics., Comment: 15 pages, 9 figures, 1 table

The spread of disinformation (maliciously spread false information) in online social networks has become an important problem in today's society. Disinformation's spread is facilitated by the fact that individuals often accept false information based on cognitive biases which predispose them to believe information that they have heard repeatedly or that aligns with their beliefs. Moreover, disinformation often spreads in direct competition with a corresponding true information. To model these phenomena, we develop a model for two competing beliefs spreading on a social network, where individuals have an internal opinion that models their cognitive biases and modulates their likelihood of adopting one of the competing beliefs. By numerical simulations of an agent-based model and a mean-field description of the dynamics, we study how the long-term dynamics of the spreading process depends on the initial conditions for the number of spreaders and the initial opinion of the population. We find that the addition of cognitive biases enriches the transient dynamics of the spreading process, facilitating behavior such as the revival of a dying belief and the overturning of an initially widespread opinion. Finally, we study how external recruitment of spreaders can lead to the eventual dominance of one of the two beliefs., Comment: 10 pages, 10 figures

Efficient and accurate prediction of physical systems is important even when the rules of those systems cannot be easily learned. Reservoir computing, a type of recurrent neural network with fixed nonlinear units, is one such prediction method and is valued for its ease of training. Organic electrochemical transistors (OECTs) are physical devices with nonlinear transient properties that can be used as the nonlinear units of a reservoir computer. We present a theoretical framework for simulating reservoir computers using OECTs as the non-linear units as a test bed for designing physical reservoir computers. We present a proof of concept demonstrating that such an implementation can accurately predict the Lorenz attractor with comparable performance to standard reservoir computer implementations. We explore the effect of operating parameters and find that the prediction performance strongly depends on the pinch-off voltage of the OECTs., Comment: 10 pages, 8 figures

We analyze the Bass model and the Susceptible-Infected (SI) model on hypergraphs with 3-body interactions. We derive the master equations for general hypernetworks, and use them to obtain explicit expressions for the expected adoption/infection level on infinite complete hypernetworks, infinite Erd\H{o}s-R\'enyi hypernetworks, and on infinite hyperlines. These expressions are exact, as they are derived without making any approximation., Comment: arXiv admin note: text overlap with arXiv:2407.10816

Motivated by the need to understand the factors driving gentrification, we introduce and analyze two simple dynamical systems that model the interplay between three potential drivers of the phenomenon. The constructed systems are based on the assumption that three canonical drivers exist: a subpopulation that increases the desirability of a neighborhood, the desirability of a neighborhood, and the average price of real estate in a neighborhood. The second model modifies the first and implements a simple rent control scheme. For both models, we investigate the linear stability of equilibria and numerically determine the characteristics of oscillatory solutions as a function of system parameters. Introducing a rent control scheme stabilizes the system, in the sense that the parameter regime under which solutions approach equilibrium is expanded. However, oscillatory time series generated by the rent control model are generally more disorganized than those generated by the non-rent control model; in fact, long-term transient chaos was observed under certain conditions in the rent control case. Our results illustrate that even simple models of urban gentrification can lead to complex temporal behavior.

Reservoir computing is a machine learning framework where the readouts from a nonlinear system (the reservoir) are trained so that the output from the reservoir, when forced with an input signal, reproduces a desired output signal. A common implementation of reservoir computers is to use a recurrent neural network as the reservoir. The design of this network can have significant effects on the performance of the reservoir computer. In this paper we study the effect of the node activation function on the ability of reservoir computers to learn and predict chaotic time series. We find that the Forecast Horizon (FH), the time during which the reservoir's predictions remain accurate, can vary by an order of magnitude across a set of 16 activation functions used in machine learning. By using different functions from this set, and by modifying their parameters, we explore whether the entropy of node activation levels or the curvature of the activation functions determine the predictive ability of the reservoirs. We find that the FH is low when the activation function is used in a region where it has low curvature, and a positive correlation between curvature and FH. For the activation functions studied we find that the largest FH generally occurs at intermediate levels of the entropy of node activation levels. Our results show that the performance of reservoir computers is very sensitive to the activation function shape. Therefore, modifying this shape in hyperparameter optimization algorithms can lead to improvements in reservoir computer performance., Comment: 12 pages, 9 figures, submitted to Journal of Physics Complexity

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 both with linear and nonlinear disturbances using food web and neuronal activity models. Finally, we discuss how to scale our method to large networks.

Identifying and suppressing unknown disturbances to dynamical systems is a problem with applications in many different fields. Here we present a model-free method to identify and suppress an unknown disturbance to an unknown system based only on previous observations of the system under the influence of a known forcing function. We find that, under very mild restrictions on the training function, our method is able to robustly identify and suppress a large class of unknown disturbances. We illustrate our scheme with the identification of both deterministic and stochastic unknown disturbances to an analog electric chaotic circuit and with numerical examples where a chaotic disturbance to various chaotic dynamical systems is identified and suppressed.

The El Farol Bar Problem is a classic computational economics problem in which agents attempt to attend a weekly event at a bar only if it is not too crowded. Each agent has access to multiple competing strategies that may be used to predict whether attendance will be above the tolerance threshold. Many different variations of the El Farol Bar Problem have been published, with a particularly broad variation in the choice of meta-strategy, the algorithm by which each agent selects the best-performing strategy. This paper discusses the varying mechanisms of self-organization within the system when different meta-strategies are used, including the existence and location of fixed points in the limit of an infinite number of agents. Informed by these mechanisms, we present an evolutionary paradigm that reduces the likelihood of degenerate cases where attendance is always too high or too low, while reducing the amplitude of fluctuations in attendance. This evolutionary paradigm sheds light on the critical role of heterogeneity of strategies selected in the emergence of a stable steady state in the El Farol Bar Problem.

The division of a social group into subgroups with opposing opinions, which we refer to as opinion disparity, is a prevalent phenomenon in society. This phenomenon has been modeled by including mechanisms such as opinion homophily, bounded confidence interactions, and social reinforcement mechanisms. In this paper we study a complementary mechanism for the formation of opinion disparity based on higher-order interactions, i.e., simultaneous interactions between multiple agents. We present an extension of the planted partition model for uniform hypergraphs as a simple model of community structure and consider the hypergraph SIS model on a hypergraph with two communities where the binary ideology can spread via links (pairwise interactions) and triangles (three-way interactions). We approximate this contagion process with a mean-field model and find that for strong enough community structure, the two communities can hold very different average opinions. We determine the regimes of structural and infectious parameters for which this opinion disparity can exist and find that the existence of these disparities is much more sensitive to the triangle community structure than to the link community structure. We show that the existence and type of opinion disparities are extremely sensitive to differences in the sizes of the two communities., Comment: 14 pages, 8 figures

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 2 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 synchronization of phase oscillators in hypergraphs, and they can be extended to hypergraphs with hyperedges of arbitrary sizes, dynamic-structural correlations, and other features., Comment: 9 pages, 5 figures

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 a 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 a stability analysis and perturbation theory.

The analysis of the dynamics of a large class of excitable systems on locally tree-like networks leads to the conclusion that at $\lambda=1$ a continuous phase transition takes place, where $\lambda$ is the largest eigenvalue of the adjacency matrix of the network. This paper is devoted to evaluate this claim for a more general case where the assumption of the linearity of the dynamical transfer function is violated with a non-linearity parameter $\beta$ which interpolates between stochastic ($\beta=0$) and deterministic ($\beta\rightarrow\infty$) dynamics. Our model shows a rich phase diagram with an absorbing state and extended critical and oscillatory regimes separated by transition and bifurcation lines which depend on the initial state. We test initial states with ($\mathbb{I}$) only one initial excited node, ($\mathbb{II}$) a fixed fraction ($10\%$) of excited nodes, for all of which the transition is of first order for $\beta>0$ with a hysteresis effect and a gap function. For the case ($\mathbb{I}$) in the thermodynamic limit the absorbing state in the only phase for all $\lambda$ values and $\beta>0$. We further develop mean-field theories for cases ($\mathbb{I}$) and ($\mathbb{II}$). For case ($\mathbb{II}$) we obtain an analytic one-dimensional map which explains the essential properties of the model, including the hysteresis diagrams and fixed points of the dynamics.

The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper we propose and explore a greedy algorithm to grow spatially embedded oscillator networks. The algorithm is constructed in such a way that nodes are sequentially added seeking to minimize the cost of the added links' length and optimize the linear stability of the growing network. We show that, for appropriate parameters, the stability of the resulting network, measured in terms of the dynamics of small perturbations and the correlation length of the disturbances, can be significantly improved with a minimal added length cost. In addition, we analyze numerically the topological properties of the resulting networks and find that, while being more stable, their degree distribution is approximately exponential and independent of the algorithm parameters. Moreover, we find that other topological parameters related with network resilience and efficiency are also affected by the proposed algorithm. Finally, we extend our findings to more general classes of networks with different sources of heterogeneity. Our results are a first step in the development of algorithms for the directed growth of oscillatory networks with desirable stability, dynamical and topological properties., Comment: 13 pages, 9 figures

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets., Comment: 11 pages, 4 figures

Topological signals defined on nodes, links and higher dimensional simplices define the dynamical state of a network or of a simplicial complex. As such, topological signals are attracting increasing attention in network theory, dynamical systems, signal processing and machine learning. Topological signals defined on the nodes are typically studied in network dynamics, while topological signals defined on links are much less explored. Here we investigate Dirac synchronization, describing locally coupled topological signals defined on the nodes and on the links of a network, and treated using the topological Dirac operator. The dynamics of signals defined on the nodes is affected by a phase lag depending on the dynamical state of nearby links and vice versa. We show that Dirac synchronization on a fully connected network is explosive with a hysteresis loop characterized by a discontinuous forward transition and a continuous backward transition. The analytical investigation of the phase diagram provides a theoretical understanding of this topological explosive synchronization. The model also displays an exotic coherent synchronized phase, also called rhythmic phase, characterized by non-stationary order parameters which can shed light on topological mechanisms for the emergence of brain rhythms., Comment: (22 pages, 10 figures)

Simplicial synchronization reveals the role that topology and geometry have in determining the dynamical properties of simplicial complexes. Simplicial network geometry and topology are naturally encoded in the spectral properties of the graph Laplacian and of the higher-order Laplacians of simplicial complexes. Here we show how the geometry of simplicial complexes induces spectral dimensions of the simplicial complex Laplacians that are responsible for changing the phase diagram of the Kuramoto model. In particular, simplicial complexes displaying a non-trivial simplicial network geometry cannot sustain a synchronized state in the infinite network limit if their spectral dimension is smaller or equal to four. This theoretical result is here verified on the Network Geometry with Flavor simplicial complex generative model displaying emergent hyperbolic geometry. On its turn simplicial topology is shown to determine the dynamical properties of the higher-order Kuramoto model. The higher-orderKuramoto model describes synchronization of topological signals, i.e. phases not only associated to the nodes of a simplicial complexes but associated also to higher-order simplices, including links, triangles and so on. This model displays discontinuous synchronization transitions when topological signals of different dimension and/or their solenoidal and irrotational projections are coupled in an adaptive way., Comment: 32 pages, 10 figures

Simplicial complexes capture the underlying network topology and geometry of complex systems ranging from the brain to social networks. Here we show that algebraic topology is a fundamental tool to capture the higher-order dynamics of simplicial complexes. In particular we consider topological signals, i.e., dynamical signals defined on simplices of different dimension, here taken to be nodes and links for simplicity. We show that coupling between signals defined on nodes and links leads to explosive topological synchronization in which phases defined on nodes synchronize simultaneously to phases defined on links at a discontinuous phase transition. We study the model on real connectomes and on simplicial complexes and network models. Finally, we provide a comprehensive theoretical approach that captures this transition on fully connected networks and on random networks treated within the annealed approximation, establishing the conditions for observing a closed hysteresis loop in the large network limit., Comment: (17 pages, 6 figures)

The analysis of games and sports as complex systems can give insights into the dynamics of human competition, and has been proven useful in soccer, basketball, and other professional sports. In this paper we present a model for dodgeball, a popular sport in US schools, and analyze it using an ordinary differential equation (ODE) compartmental model and stochastic agent-based game simulations. The ODE model reveals a rich landscape with different game dynamics occurring depending on the strategies used by the teams, which can in some cases be mapped to scenarios in competitive species models. Stochastic agent-based game simulations confirm and complement the predictions of the deterministic ODE models. In some scenarios, game victory can be interpreted as a noise-driven escape from the basin of attraction of a stable fixed point, resulting in extremely long games when the number of players is large. Using the ODE and agent-based models, we construct a strategy to increase the probability of winning., Comment: Added Acknowledgments

The dynamics of network social contagion processes such as opinion formation and epidemic spreading are often mediated by interactions between multiple nodes. Previous results have shown that these higher-order interactions can profoundly modify the dynamics of contagion processes, resulting in bistability, hysteresis, and explosive transitions. In this paper, we present and analyze a hyperdegree-based mean-field description of the dynamics of the SIS model on hypergraphs, i.e. networks with higher-order interactions, and illustrate its applicability with the example of a hypergraph where contagion is mediated by both links (pairwise interactions) and triangles (three-way interactions). We consider various models for the organization of link and triangle structure, and different mechanisms of higher-order contagion and healing. We find that explosive transitions can be suppressed by heterogeneity in the link degree distribution, when links and triangles are chosen independently, or when link and triangle connections are positively correlated when compared to the uncorrelated case. We verify these results with microscopic simulations of the contagion process and with analytic predictions derived from the mean-field model. Our results show that the structure of higher-order interactions can have important effects on contagion processes on hypergraphs., Comment: 14 pages, 10 figures

Experiments suggest that cerebral cortex gains several functional advantages by operating in a dynamical regime near the critical point of a phase transition. However, a long-standing criticism of this hypothesis is that critical dynamics are rather noisy, which might be detrimental to aspects of brain function that require precision. If the cortex does operate near criticality, how might it mitigate the noisy fluctuations? One possibility is that other parts of the brain may act to control the fluctuations and reduce cortical noise. To better understand this possibility, here we numerically and analytically study a network of binary neurons. We determine how efficacy of controlling the population firing rate depends on proximity to criticality as well as different structural properties of the network. We found that control is most effective - errors are minimal for the widest range of target firing rates - near criticality. Optimal control is slightly away from criticality for networks with heterogeneous degree distributions. Thus, while criticality is the noisiest dynamical regime, it is also the regime that is easiest to control, which may offer a way to mitigate the noise., Comment: 5 pages, 3 figures

We introduce and test a general machine-learning-based technique for the inference of short term causal dependence between state variables of an unknown dynamical system from time series measurements of its state variables. Our technique leverages the results of a machine learning process for short time prediction to achieve our goal. The basic idea is to use the machine learning to estimate the elements of the Jacobian matrix of the dynamical flow along an orbit. The type of machine learning that we employ is reservoir computing. We present numerical tests on link inference of a network of interacting dynamical nodes. It is seen that dynamical noise can greatly enhance the effectiveness of our technique, while observational noise degrades the effectiveness. We believe that the competition between these two opposing types of noise will be the key factor determining the success of causal inference in many of the most important application situations., Comment: 28 pages, 4 Figures, Accepted in Chaos

Community structure and interaction delays are common features of ensembles of network coupled oscillators, but their combined effect on the emergence of synchronization has not been studied in detail. We study the transitions between macroscopic states in coupled oscillator systems with community structure and time delays. We show that the combination of these two properties gives rise to non-monotonic transitions, whereby increasing the global coupling strength can both inhibit and promote synchronization, yielding both desynchronization and synchronization transitions. For relatively wide parameter choices we also observe asymmetric suppression of synchronization, where communities compete to suppress one another's synchronization properties until one or more win, totally suppressing the others to effective incoherence. Using the ansatz of Ott and Antonsen we provide analytical descriptions for these transitions that confirm numerical simulations.

We investigate the emergence of a time crystal in a driven-dissipative many-body spin array. In this system the interplay between incoherent spin pumping and collective emission stabilizes a synchronized non-equilibrium steady state which in the thermodynamic limit features a self-generated time-periodic pattern imposed by collective elastic interactions. In contrast to prior realizations where the time symmetry is already broken by an external drive, here it is only spontaneously broken by the elastic exchange interactions and manifest in the two-time correlation spectrum. Employing a combination of exact numerical calculations and a second-order cumulant expansion, we investigate the impact of many-body correlations on the time crystal formation and establish a connection between the regime where it is stable and a slow growth rate of the mutual information, signalling that the time crystal studied here is an emergent semi-classical out-of-equilibrium state of matter. We also confirm the rigidity of the time crystal to single-particle dephasing. Finally, we discuss an experimental implementation using long-lived dipoles in an optical cavity., Comment: v1: Initial commit; v2: Added references, fixed a couple typos, and made some small, stylistic changes; v3: Update to reflect publication. Includes additional references and some minor additions

It is widely appreciated that well-balanced excitation and inhibition are necessary for proper function in neural networks. However, in principle, such balance could be achieved by many possible configurations of excitatory and inhibitory strengths, and relative numbers of excitatory and inhibitory neurons. For instance, a given level of excitation could be balanced by either numerous inhibitory neurons with weak synapses, or few inhibitory neurons with strong synapses. Among the continuum of different but balanced configurations, why should any particular configuration be favored? Here we address this question in the context of the entropy of network dynamics by studying an analytically tractable network of binary neurons. We find that entropy is highest at the boundary between excitation-dominant and inhibition-dominant regimes. Entropy also varies along this boundary with a trade-off between high and robust entropy: weak synapse strengths yield high network entropy which is fragile to parameter variations, while strong synapse strengths yield a lower, but more robust, network entropy. In the case where inhibitory and excitatory synapses are constrained to have similar strength, we find that a small, but non-zero fraction of inhibitory neurons, like that seen in mammalian cortex, results in robust and relatively high entropy.

Various functions of a network of excitable units can be enhanced if the network is in the `critical regime', where excitations are, on average, neither damped nor amplified. An important question is how can such networks self-organize to operate in the critical regime. Previously it was shown that regulation via resource transport on a secondary network can robustly maintain the primary network dynamics in a balanced state where activity doesn't grow or decay. Here we show that this inter-network regulation process robustly produces a power-law distribution of activity avalanches, as observed in experiments, over ranges of model parameters spanning orders of magnitude. We also show that the resource transport over the secondary network protects the system against the destabilizing effect of local variations in parameters and heterogeneity in network structure. For homogeneous networks, we derive a reduced 3-dimensional map which reproduces the behavior of the full system., Comment: 6 pages, 5 figures

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.

Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda_i\}$ and eigenvectors $\{{\bf u}_i\}$ of a covariance matrix are central to such endeavors, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda}_i\}$ and eigenvectors $\{\tilde{{\bf u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyze eigenvector error $\|{\bf u}_i - \tilde{{\bf u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties---particularly, we assume the distribution of population eigenvalues weakly converges as $p\to\infty$ to a spectral density $\rho(\lambda)$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda$ and approximate the distribution of the expected square error $r= \mathbb{E}\left[\| {\bf u}_i - \tilde{{\bf u}}_i \|^2\right]$ across the matrix ensemble for all ${\bf u}_i$ associated with $\lambda_i=\lambda$. We find, for example, that for sufficiently large matrix size $p$ and sample size $n>p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that eigenvector error is extremely heterogeneous---even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments., Comment: 24 pages, 8 figures

Learning and memory are acquired through long-lasting changes in synapses. In the simplest models, such synaptic potentiation typically leads to runaway excitation, but in reality there must exist processes that robustly preserve overall stability of the neural system dynamics. How is this accomplished? Various approaches to this basic question have been considered. Here we propose a particularly compelling and natural mechanism for preserving stability of learning neural systems. This mechanism is based on the global processes by which metabolic resources are distributed to the neurons by glial cells. Specifically, we introduce and study a model comprised of two interacting networks: a model neural network interconnected by synapses which undergo spike-timing dependent plasticity (STDP); and a model glial network interconnected by gap junctions which diffusively transport metabolic resources among the glia and, ultimately, to neural synapses where they are consumed. Our main result is that the biophysical constraints imposed by diffusive transport of metabolic resources through the glial network can prevent runaway growth of synaptic strength, both during ongoing activity and during learning. Our findings suggest a previously unappreciated role for glial transport of metabolites in the feedback control stabilization of neural network dynamics during learning., Comment: 8 pages, 5 figures

The Hamiltonian Mean Field (HMF) 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 occurs at a coupling constant $K$ inversely proportional to the largest eigenvalue of the adjacency matrix. We derive a closed system of equations for a set of local order parameters and use these equations to study the effect of network heterogeneity on the synchronization of the rotors. We find that for values of $K$ just beyond the transition to synchronization the degree of synchronization is highly dependent on the network's heterogeneity, but that for large values of $K$ the degree of synchronization is robust to changes in the heterogeneity of the network's degree distribution. Our results are illustrated with numerical simulations on Erd\"os-Renyi networks and networks with power-law degree distributions., Comment: 14 pages, 10 figures, 1 Appendix

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.

Macroscopic ensembles of radiating dipoles are ubiquitous in the physical and natural sciences. In the classical limit the dipoles can be described as damped-driven oscillators, which are able to spontaneously synchronize and collectively lock their phases. Here we investigate the correspond- ing phenomenon in the quantum regime with arrays of quantized two-level systems coupled via long-range and anisotropic dipolar interactions. Our calculations demonstrate that the dipoles may overcome the decoherence induced by quantum fluctuations and inhomogeneous couplings and evolve to a synchronized steady-state. This steady-state bears much similarity to that observed in classical systems, and yet also exhibits genuine quantum properties such as quantum correlations and quan- tum phase diffusion (reminiscent of lasing). Our predictions could be relevant for the development of better atomic clocks and a variety of noise tolerant quantum devices.

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.

Employing the Kuramoto model as an illustrative example, we show how the use of the mean field approximation can be applied to large networks of phase oscillators with assortativity. We then use the ansatz of Ott and Antonsen [Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a system of ordinary differential equations. The resulting formulation is illustrated by application to a network Kuramoto problem with degree assortativity and correlation between the node degrees and the natural oscillation frequencies. Good agreement is found between the solutions of the reduced set of ordinary differential equations obtained from our theory and full simulations of the system. These results highlight the ability of our method to capture all the phase transitions (bifurcations) and system attractors. One interesting result is that degree assortativity can induce transitions from a steady macroscopic state to a temporally oscillating macroscopic state through both (presumed) Hopf and SNIPER (saddle-node, infinite period) bifurcations. Possible use of these techniques to a broad class of phase oscillator network problems is discussed., Comment: 8 pages, 7 figures

In this paper, a comprehensive study of the the downlink performance in a heterogeneous cellular network (or hetnet) is conducted. A general hetnet model is considered consisting of an arbitrary number of open-access and closed-access tier of base stations (BSs) arranged according to independent homogeneous Poisson point processes. The BSs of each tier have a constant transmission power, random fading coefficient with an arbitrary distribution and arbitrary path-loss exponent of the power-law path-loss model. For such a system, analytical characterizations for the coverage probability and average rate at an arbitrary mobile-station (MS), and average per-tier load are derived for both the max-SINR connectivity and nearest-BS connectivity models. Using stochastic ordering, interesting properties and simplifications for the hetnet downlink performance are derived by relating these two connectivity models to the maximum instantaneous received power (MIRP) connectivity model and the maximum biased received power (MBRP) connectivity models, respectively, providing good insights about the hetnets and the downlink performance in these complex networks. Furthermore, the results also demonstrate the effectiveness and analytical tractability of the stochastic geometric approach to study the hetnet performance., Comment: 13 pages, 3 figures, 1 table, to be submitted to Transactions on Wireless Communications

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 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 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 sub-linearly with this length scale. Those results are also shown to be robust against cell-to-cell fluctuations owing 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., Comment: 26 pages, 26 figures

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., Comment: 5 pages, 5 figures

The collective dynamics of a network of excitable nodes changes dramatically when inhibitory nodes are introduced. We consider inhibitory nodes which may be activated just like excitatory nodes but, upon activating, decrease the probability of activation of network neighbors. We show that, although the direct effect of inhibitory nodes is to decrease activity, the collective dynamics becomes self-sustaining. We explain this counterintuitive result by defining and analyzing a "branching function" which may be thought of as an activity-dependent branching ratio. The shape of the branching function implies that for a range of global coupling parameters dynamics are self-sustaining. Within the self-sustaining region of parameter space lies a critical line along which dynamics take the form of avalanches with universal scaling of size and duration, embedded in ceaseless timeseries of activity. Our analyses, confirmed by numerical simulation, suggest that inhibition may play a counterintuitive role in excitable networks., Comment: 11 pages, 6 figures

While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations., Comment: 12 pages, 9 figures

We introduce a model to study the effect of degree-frequency correlations on synchronization in networks of coupled oscillators. Analyzing this model, we find several remarkable characteristics. We find a stationary synchronized state that is (i) universal, i.e., the degree of synchrony, as measured by a global order parameter, is independent of network topology, and (ii) fully phase-locked, i.e., all oscillators become simultaneously phase-locked despite having different natural frequencies. This state separates qualitatively different behaviors for two other classes of correlations where, respectively, slow and fast oscillators can remain unsynchronized. We close by presenting analysis of the dynamics under arbitrary degree-frequency correlations., Comment: 6 pages, 5 figures

Using recent dimensionality reduction techniques in large systems of coupled phase oscillators exhibiting bistability, we analyze complex macroscopic behavior arising when the coupling between oscillators is allowed to evolve slowly as a function of either macroscopic or local system properties. For example, we observe macroscopic excitability and intermittent synchrony in a system of time-delayed Kuramoto oscillators with Hebbian and anti-Hebbian learning. We demonstrate the robustness of our findings by considering systems with increasing complexity, including time-delayed oscillators with adaptive network structure and community interaction, as well as a system with bimodally distributed frequencies.

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 SINR under the open access cell association scheme. For such a general setting, we provide an analytical characterization of the coverage probability at the MS., Comment: 6 pages, 5 figures, submitted to IEEE Globecom 2012 - Wireless Communications Symposium on Apr 2, 2012

We study synchronization of Kuramoto oscillators in strongly modular networks in which the structure of the network inside each community is averaged. We find that the dynamics of the interacting communities can be described as an ensemble of coupled planar oscillators. In the limit of a large number of communities, we find a low dimensional description of the level of synchronization between the communities. In this limit, we describe bifurcations between incoherence, local synchrony, and global synchrony. We compare the predictions of this simplified model with simulations of heterogeneous networks in which the internal structure of each community is preserved and find excellent agreement. Finally, we investigate synchronization in networks where several layers of communities within communities may be present.

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 firings of the nodes to which it is connected, resulting in a cascade of firings. This type of process is relevant to a wide variety of situations, including neuroscience, cascading failures on electrical power grids, and epidemology. We find that the statistics of avalanches can be characterized in terms of the largest eigenvalue and corresponding eigenvector of an appropriate adjacency matrix which 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., Comment: 11 pages, 7 figures

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 nodal lines separating discordant regions. We show that this jump unidirectionally pins nodal lines 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 nodal line motion that is strongly arrhythmogenic., Comment: 4 pages, 3 figures

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, 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.