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

Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions

Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.Comment: 7 pages, 4 figure

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

Breaking Synchrony by Heterogeneity in Complex Networks

For networks of pulse-coupled oscillators with complex connectivity, we demonstrate that in the presence of coupling heterogeneity precisely timed periodic firing patterns replace the state of global synchrony that exists in homogenous networks only. With increasing disorder, these patterns persist until they reach a critical temporal extent that is of the order of the interaction delay. For stronger disorder these patterns cease to exist and only asynchronous, aperiodic states are observed. We derive self-consistency equations to predict the precise temporal structure of a pattern from the network heterogeneity. Moreover, we show how to design heterogenous coupling architectures to create an arbitrary prescribed pattern.Comment: 4 pages, 3 figure

Topological Speed Limits to Network Synchronization

We study collective synchronization of pulse-coupled oscillators interacting on asymmetric random networks. We demonstrate that random matrix theory can be used to accurately predict the speed of synchronization in such networks in dependence on the dynamical and network parameters. Furthermore, we show that the speed of synchronization is limited by the network connectivity and stays finite, even if the coupling strength becomes infinite. In addition, our results indicate that synchrony is robust under structural perturbations of the network dynamics.Comment: 5 pages, 3 figure

Network representations of non-equilibrium steady states: Cycle decompositions, symmetries and dominant paths

Non-equilibrium steady states (NESS) of Markov processes give rise to non-trivial cyclic probability fluxes. Cycle decompositions of the steady state offer an effective description of such fluxes. Here, we present an iterative cycle decomposition exhibiting a natural dynamics on the space of cycles that satisfies detailed balance. Expectation values of observables can be expressed as cycle "averages", resembling the cycle representation of expectation values in dynamical systems. We illustrate our approach in terms of an analogy to a simple model of mass transit dynamics. Symmetries are reflected in our approach by a reduction of the minimal number of cycles needed in the decomposition. These features are demonstrated by discussing a variant of an asymmetric exclusion process (TASEP). Intriguingly, a continuous change of dominant flow paths in the network results in a change of the structure of cycles as well as in discontinuous jumps in cycle weights.Comment: 3 figures, 4 table

The role of Nrf1 and Nrf2 in the regulation of glutathione and redox dynamics in the developing zebrafish embryo

© The Author(s), 2017. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Redox Biology 13 (2017): 207–218, doi:10.1016/j.redox.2017.05.023.Redox signaling is important for embryogenesis, guiding pathways that govern processes crucial for embryo patterning, including cell polarization, proliferation, and apoptosis. Exposure to pro-oxidants during this period can be deleterious, resulting in altered physiology, teratogenesis, later-life diseases, or lethality. We previously reported that the glutathione antioxidant defense system becomes increasingly robust, including a doubling of total glutathione and dynamic shifts in the glutathione redox potential at specific stages during embryonic development in the zebrafish, Danio rerio. However, the mechanisms underlying these changes are unclear, as is the effectiveness of the glutathione system in ameliorating oxidative insults to the embryo at different stages. Here, we examine how the glutathione system responds to the model pro-oxidants tert-butylhydroperoxide and tert-butylhydroquinone at different developmental stages, and the role of Nuclear factor erythroid 2-related factor (Nrf) proteins in regulating developmental glutathione redox status. Embryos became increasingly sensitive to pro-oxidants after 72 h post-fertilization (hpf), after which the duration of the recovery period for the glutathione redox potential was increased. To determine whether the doubling of glutathione or the dynamic changes in glutathione redox potential are mediated by zebrafish paralogs of Nrf transcription factors, morpholino oligonucleotides were used to knock down translation of Nrf1 and Nrf2 (nrf1a, nrf1b, nrf2a, nrf2b). Knockdown of Nrf1a or Nrf1b perturbed glutathione redox state until 72 hpf. Knockdown of Nrf2 paralogs also perturbed glutathione redox state but did not significantly affect the response of glutathione to pro-oxidants. Nrf1b morphants had decreased gene expression of glutathione synthesis enzymes, while hsp70 increased in Nrf2b morphants. This work demonstrates that despite having a more robust glutathione system, embryos become more sensitive to oxidative stress later in development, and that neither Nrf1 nor Nrf2 alone appear to be essential for the response and recovery of glutathione to oxidative insults.This research was supported by several NIH grants, including F32ES028085 (to KES), F32ES017585 (to ART-L), F32ES019832 (to LMW), P20GM103423 (to LMW), R01ES025748 (to ART-L), R01ES015912 (JJS), and R01ES016366 (MEH). Additional research support was provided by the J. Seward Johnson Fund at WHOI and the WHOI Postdoctoral Scholar Award with funding from Walter A. and Hope Noyes Smith (to ART-L)