Does dynamics reflect topology in directed networks?

We present and analyze a topologically induced transition from ordered, synchronized to disordered dynamics in directed networks of oscillators. The analysis reveals where in the space of networks this transition occurs and its underlying mechanisms. If disordered, the dynamics of the units is precisely determined by the topology of the network and thus characteristic for it. We develop a method to predict the disordered dynamics from topology. The results suggest a new route towards understanding how the precise dynamics of the units of a directed network may encode information about its topology.Comment: 7 pages, 4 figures, Europhysics Letters, accepte

Unstable Attractors: Existence and Robustness in Networks of Oscillators With Delayed Pulse Coupling

We consider unstable attractors; Milnor attractors $A$ such that, for some neighbourhood $U$ of $A$, almost all initial conditions leave $U$. Previous research strongly suggests that unstable attractors exist and even occur robustly (i.e. for open sets of parameter values) in a system modelling biological phenomena, namely in globally coupled oscillators with delayed pulse interactions. In the first part of this paper we give a rigorous definition of unstable attractors for general dynamical systems. We classify unstable attractors into two types, depending on whether or not there is a neighbourhood of the attractor that intersects the basin in a set of positive measure. We give examples of both types of unstable attractor; these examples have non-invertible dynamics that collapse certain open sets onto stable manifolds of saddle orbits. In the second part we give the first rigorous demonstration of existence and robust occurrence of unstable attractors in a network of oscillators with delayed pulse coupling. Although such systems are technically hybrid systems of delay differential equations with discontinuous `firing' events, we show that their dynamics reduces to a finite dimensional hybrid system system after a finite time and hence we can discuss Milnor attractors for this reduced finite dimensional system. We prove that for an open set of phase resetting functions there are saddle periodic orbits that are unstable attractors.Comment: 29 pages, 8 figures,submitted to Nonlinearit

Speed of synchronization in complex networks of neural oscillators Analytic results based on Random Matrix Theory

We analyze the dynamics of networks of spiking neural oscillators. First, we present an exact linear stability theory of the synchronous state for networks of arbitrary connectivity. For general neuron rise functions, stability is determined by multiple operators, for which standard analysis is not suitable. We describe a general non-standard solution to the multi-operator problem. Subsequently, we derive a class of rise functions for which all stability operators become degenerate and standard eigenvalue analysis becomes a suitable tool. Interestingly, this class is found to consist of networks of leaky integrate and fire neurons. For random networks of inhibitory integrate-and-fire neurons, we then develop an analytical approach, based on the theory of random matrices, to precisely determine the eigenvalue distribution. This yields the asymptotic relaxation time for perturbations to the synchronous state which provides the characteristic time scale on which neurons can coordinate their activity in such networks. For networks with finite in-degree, i.e. finite number of presynaptic inputs per neuron, we find a speed limit to coordinating spiking activity: Even with arbitrarily strong interaction strengths neurons cannot synchronize faster than at a certain maximal speed determined by the typical in-degree.Comment: 17 pages, 12 figures, submitted to Chao

Chaos in Symmetric Phase Oscillator Networks

Phase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatory units in physics and biology. The order parameter which quantifies synchronization was so far found to be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems of identical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating order parameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary to obtain chaos, i.e., nonlinear interactions of phases give rise to the necessary instabilities.Comment: 4 pages; Accepted by Physical Review Letter

Stochastic facilitation in heteroclinic communication channels

Biological neural systems encode and transmit information as patterns of activity tracing complex trajectories in high-dimensional state spaces, inspiring alternative paradigms of information processing. Heteroclinic networks, naturally emerging in artificial neural systems, are networks of saddles in state space that provide a transparent approach to generate complex trajectories via controlled switches among interconnected saddles. External signals induce specific switching sequences, thus dynamically encoding inputs as trajectories. Recent works have focused either on computational aspects of heteroclinic networks, i.e., Heteroclinic Computing, or their stochastic properties under noise. Yet, how well such systems may transmit information remains an open question. Here, we investigate the information transmission properties of heteroclinic networks, studying them as communication channels. Choosing a tractable but representative system exhibiting a heteroclinic network, we investigate the mutual information rate (MIR) between input signals and the resulting sequences of states as the level of noise varies. Intriguingly, MIR does not decrease monotonically with increasing noise. Intermediate noise levels indeed maximize the information transmission capacity by promoting an increased yet controlled exploration of the underlying network of states. Complementing standard stochastic resonance, these results highlight the constructive effect of stochastic facilitation (i.e., noise-enhanced information transfer) on heteroclinic communication channels and possibly on more general dynamical systems exhibiting complex trajectories in state space

Ovarian Tumors in Children and Adolescents

Ovarian tumors are uncommon in the child and adolescent population, but given that cancer is a leading cause of death even in this age group it is important for the general practice physician to be aware of the signs and symptoms. These can include abdominal pain, increased abdominal girth, a palpable mass on pelvic or abdominal exam, nausea and vomiting, decreased appetite, weight loss, constipation, back pain, leg cramps, dyspareunia, urinary symptoms, abnormal uterine bleeding, and ascites, to name a few. The types of gynecological tumors seen in adults are also seen in children except in differing frequencies. The most common type of ovarian tumor in children is the germ cell tumor, of which there are several subtypes including dermoid cysts, dysgerminoma, yolk sac tumor, immature teratomas, and embryonal carcinomas

Data-driven model of the power-grid frequency dynamics

The energy system is rapidly changing to accommodate the increasing number of renewable generators and the general transition towards a more sustainable future. Simultaneously, business models and market designs evolve, affecting power-grid operation and power-grid frequency. Problems raised by this ongoing transition are increasingly addressed by transdisciplinary research approaches, ranging from purely mathematical modelling to applied case studies. These approaches require a stochastic description of consumer behaviour, fluctuations by renewables, market rules, and how they influence the stability of the power-grid frequency. Here, we introduce an easy-to-use, data-driven, stochastic model for the power-grid frequency and demonstrate how it reproduces key characteristics of the observed statistics of the Continental European and British power grids. Using data analysis tools and a Fokker–Planck approach, we estimate parameters of our deterministic and stochastic model. We offer executable code and guidelines on how to use the model on any power grid for various mathematical or engineering applications