Isochronous Dynamics in Pulse Coupled Oscillator Networks with Delay

We consider a network of identical pulse-coupled oscillators with delay and all-to-all coupling. We demonstrate that the discontinuous nature of the dynamics induces the appearance of isochronous regions---subsets of the phase space filled with periodic orbits having the same period. For fixed values of the network parameters each such isochronous region corresponds to a subset of initial states on an appropriate surface of section with non-zero dimension such that all periodic orbits in this set have qualitatively similar dynamical behaviour. We analytically and numerically study in detail such an isochronous region, give a proof of its existence, and describe its properties. We further describe other isochronous regions that appear in the system

Proportional-Integral Synchronisation for Non-identical Wireless Packet-Coupled Oscillators with Delays

Precise timing among wireless sensor nodes is a key enabling technology for time-sensitive industrial Wireless Sensor Networks (WSNs). However, the accuracy of timing is degraded by manufacturing tolerance, ageing of crystal oscillators, and communication delays. This paper develops a framework of Packet-Coupled Oscillator (PkCOs) to characterise the dynamics of communication and time synchronisation of clocks in WSNs. A non-identical clock is derived to describe the embedded clock's behaviour accurately. The Proportional-Integral (PI) packet coupling scheme is proposed for synchronising networked embedded clocks, meanwhile, scheduling wireless Sync packets to different slots for transmission. It also possesses the feature of automatically eliminating the effects of unknown processing delay, which further improves synchronisation performance. The rigorous theoretical analysis of PI-based PkCOs is presented via studying a closed-loop time synchronisation system. The performance of PI-based PkCOs is evaluated on a hardware testbed of IEEE 802.15.4 WSN. The experimental results show that the precision of the proportional-integral PkCOs protocol is as high as 60us (i.e., 2 ticks) for 32.768kHz crystal oscillator-based clocks

Pulse shape and voltage-dependent synchronization in spiking neuron networks

Pulse-coupled spiking neural networks are a powerful tool to gain mechanistic insights into how neurons self-organize to produce coherent collective behavior. These networks use simple spiking neuron models, such as the $\theta$-neuron or the quadratic integrate-and-fire (QIF) neuron, that replicate the essential features of real neural dynamics. Interactions between neurons are modeled with infinitely narrow pulses, or spikes, rather than the more complex dynamics of real synapses. To make these networks biologically more plausible, it has been proposed that they must also account for the finite width of the pulses, which can have a significant impact on the network dynamics. However, the derivation and interpretation of these pulses is contradictory and the impact of the pulse shape on the network dynamics is largely unexplored. Here, I take a comprehensive approach to pulse-coupling in networks of QIF and $\theta$-neurons. I argue that narrow pulses activate voltage-dependent synaptic conductances and show how to implement them in QIF neurons such that their effect can last through the phase after the spike. Using an exact low-dimensional description for networks of globally coupled spiking neurons, I prove for instantaneous interactions that collective oscillations emerge due to an effective coupling through the mean voltage. I analyze the impact of the pulse shape by means of a family of smooth pulse functions with arbitrary finite width and symmetric or asymmetric shapes. For symmetric pulses, the resulting voltage-coupling is little effective in synchronizing neurons, but pulses that are slightly skewed to the phase after the spike readily generate collective oscillations. The results unveil a voltage-dependent spike synchronization mechanism in neural networks, which is facilitated by pulses of finite width and complementary to traditional synaptic transmission.Comment: 38 pages, 11 figure

Synchronization in periodically driven and coupled stochastic systems-A discrete state approach

Wir untersuchen das Verhalten von stochastischen bistabilen und erregbaren Systemen auf der Basis einer Modellierung mit diskreten Zuständen. In Ergänzung zum bekannten Markovschen Zwei-Zustandsmodell bistabiler stochastischer Dynamik stellen wir ein nicht Markovsches Drei-Zustandsmodell für erregbare Systeme vor. Seine relative Einfachheit, verglichen mit stochastischen Modellen erregbarer Dynamik mit kontinuierlichem Phasenraum, ermöglicht eine teilweise analytische Auswertung in verschiedenen Zusammenhängen. Zunächst untersuchen wir den gemeinsamen Einfluß eines periodischen Treibens und Rauschens. Dieser wird entweder mit Hilfe spektraler Größen oder durch Synchronisation des Systems mit dem treibenden Signal charakterisiert. Wir leiten analytische Ausdrücke für die spektrale Leistungsverstärkung und das Signal-zu-Rauschen Verhältnis für periodisch getriebene Renewal-Prozesse her und wenden diese auf das diskrete Modell für erregbare Dynamik an. Stochastische Synchronization des Systems mit dem treibenden Signal wird auf der Basis der Diffusionseigenschaften der Übergangsereignisse zwischen den diskreten Zuständen untersucht. Wir leiten allgemeine Formeln her, um die mittlere Häufigkeit dieser Ereignisse sowie deren effektiven Diffusionskoeffizienten zu berechnen. Über die konkrete Anwendung auf die untersuchten diskreten Modelle hinaus stellen diese Ergebnisse ein neues Werkzeug für die Untersuchung periodischer Renewal-Prozesse dar. Schließlich betrachten wir noch das Verhalten global gekoppelter bistabiler und erregbarer Systeme. Im Gegensatz zu bistabilen System können erregbare Systeme synchronisiert werden und zeigen kohärente Oszillationen. Alle Untersuchungen des nicht Markovschen Drei-Zustandsmodells werden mit dem prototypischen Modell für erregbare Dynamik, dem FitzHugh-Nagumo System, verglichen und zeigen eine gute Übereinstimmung.We investigate the behavior of stochastic bistable and excitable dynamics based on a discrete state modeling. In addition to the well known Markovian two state model for bistable dynamics we introduce a non Markovian three state model for excitable systems. Its relative simplicity compared to stochastic models of excitable dynamics with continuous phase space allows to obtain analytical results in different contexts. First, we study the joint influence of periodic signals and noise, both based on a characterization in terms of spectral quantities and in terms of synchronization with the periodic driving. We present expressions for the spectral power amplification and signal to noise ratio for renewal processes driven by periodic signals and apply these results to the discrete model for excitable systems. Stochastic synchronization of the system to the driving signal is investigated based on diffusion properties of the transition events between the discrete states. We derive general results for the mean frequency and effective diffusion coefficient which, beyond the application to the discrete models considered in this work, provide a new tool in the study of periodically driven renewal processes. Finally the behavior of globally coupled excitable and bistable units is investigated based on the discrete state description. In contrast to the bistable systems, the excitable system exhibits synchronization and thus coherent oscillations. All investigations of the non Markovian three state model are compared with the prototypical continuous model for excitable dynamics, the FitzHugh-Nagumo system, revealing a good agreement between both models

Networked Dynamical Systems: Privacy, Control, and Cognition

Many natural and man-made systems, ranging from thenervous system to power and transportation grids to societies, exhibitdynamic behaviors that evolve over a sparse and complex network. This networked aspect raises significant challenges and opportunities for the identification, analysis, and control of such dynamic behaviors. While some of these challenges emanate from the networked aspect \emph{per se} (such as the sparsity of connections between system components and the interplay between nodal \emph{communication} and network dynamics), various challenges arise from the specific application areas (such as privacy concerns in cyber-physical systems or the need for \emph{scalable} algorithm designs due to the large size of various biological and engineered networks). On the other hand, networked systems provide significant opportunities and allow for performance and robustness levels that are far beyond reach for centralized systems, with examples ranging from the Internet (of Things) to the smart grid and the brain. This dissertation aims to address several of these challenges and harness these opportunities. The dissertation is divided into three parts. In the first part, we study privacy concerns whose resolution is vital for the utility of networked cyber-physical systems. We study the problems of average consensus and convex optimization as two principal distributed computations occurring over networks and design algorithm with rigorous privacy guarantees that provide a \emph{best achievable} tradeoff between network utility and privacy. In the second part, we analyze networks with resource constraints. More specifically, we study three problems of stabilization under communication (bandwidth and latency) limitations in sensing and actuation, optimal time-varying control scheduling problem under limited number of actuators and control energy, and the structure identification problem of under-sensed networks (i.e., networks with latent nodes). Finally in the last part, we focus on the intersection of networked dynamical systems and neuroscience and draw connections between brain network dynamics and two extensively studied but yet not fully understood neuro-cognitive phenomena: goal-driven selective attention and neural oscillations. Using a novel axiomatic approach, we establish these connections in the form of necessary and/or sufficient conditions on the network structure that match the network output trajectories with experimentally observed brain activity

Complex Systems: Nonlinearity and Structural Complexity in spatially extended and discrete systems

Resumen Esta Tesis doctoral aborda el estudio de sistemas de muchos elementos (sistemas discretos) interactuantes. La fenomenología presente en estos sistemas esta dada por la presencia de dos ingredientes fundamentales: (i) Complejidad dinámica: Las ecuaciones del movimiento que rigen la evolución de los constituyentes son no lineales de manera que raramente podremos encontrar soluciones analíticas. En el espacio de fases de estos sistemas pueden coexistir diferentes tipos de trayectorias dinámicas (multiestabilidad) y su topología puede variar enormemente dependiendo de dos parámetros usados en las ecuaciones. La conjunción de dinámica no lineal y sistemas de muchos grados de libertad (como los que aquí se estudian) da lugar a propiedades emergentes como la existencia de soluciones localizadas en el espacio, sincronización, caos espacio-temporal, formación de patrones, etc... (ii) Complejidad estructural: Se refiere a la existencia de un alto grado de aleatoriedad en el patrón de las interacciones entre los componentes. En la mayoría de los sistemas estudiados esta aleatoriedad se presenta de forma que la descripción de la influencia del entorno sobre un único elemento del sistema no puede describirse mediante una aproximación de campo medio. El estudio de estos dos ingredientes en sistemas extendidos se realizará de forma separada (Partes I y II de esta Tesis) y conjunta (Parte III). Si bien en los dos primeros casos la fenomenología introducida por cada fuente de complejidad viene siendo objeto de amplios estudios independientes a lo largo de los últimos años, la conjunción de ambas da lugar a un campo abierto y enormemente prometedor, donde la interdisciplinariedad concerniente a los campos de aplicación implica un amplio esfuerzo de diversas comunidades científicas. En particular, este es el caso del estudio de la dinámica en sistemas biológicos cuyo análisis es difícil de abordar con técnicas exclusivas de la Bioquímica, la Física Estadística o la Física Matemática. En definitiva, el objetivo marcado en esta Tesis es estudiar por separado dos fuentes de complejidad inherentes a muchos sistemas de interés para, finalmente, estar en disposición de atacar con nuevas perspectivas problemas relevantes para la Física de procesos celulares, la Neurociencia, Dinámica Evolutiva, etc..