This book brings together two emerging research areas: synchronization in coupled nonlinear systems and complex networks, and study conditions under which a complex network of dynamical systems synchronizes. While there are many texts that study synchronization in chaotic systems or properties of complex networks, there are few texts that consider the intersection of these two very active and interdisciplinary research areas. The main theme of this book is that synchronization conditions can be related to graph theoretical properties of the underlying coupling topology. The book introduces ideas from systems theory, linear algebra and graph theory and the synergy between them that are necessary to derive synchronization conditions. Many of the results, which have been obtained fairly recently and have until now not appeared in textbook form, are presented with complete proofs. This text is suitable for graduate-level study or for researchers who would like to be better acquainted with the latest research in this area. Sample Chapter(s). Chapter 1: Introduction (76 KB). Contents: Graphs, Networks, Laplacian Matrices and Algebraic Connectivity; Graph Models; Synchronization in Networks of Nonlinear Continuous-Time Dynamical Systems; Synchronization in Networks of Coupled Discrete-Time Systems; Synchronization in Network of Systems with Linear Dynamics; Agreement and Consensus Problems in Groups of Interacting Agents. Readership: Graduate students and researchers in physics, applied mathematics and engineering.

Network science offers a powerful language to represent and study complex systems composed of interacting elements — from the Internet to social and biological systems. In its standard formulation, this framework relies on the assumption that the underlying topology is static, or changing very slowly as compared to dynamical processes taking place on it, e.g., epidemic spreading or navigation. Fuelled by the increasing availability of longitudinal networked data, recent empirical observations have shown that this assumption is not valid in a variety of situations. Instead, often the network itself presents rich temporal properties and new tools are required to properly describe and analyse their behaviour.A Guide to Temporal Networks presents recent theoretical and modelling progress in the emerging field of temporally varying networks, and provides connections between different areas of knowledge required to address this multi-disciplinary subject. After an introduction to key concepts on networks and stochastic dynamics, the authors guide the reader through a coherent selection of mathematical and computational tools for network dynamics. Perfect for students and professionals, this book is a gateway to an active field of research developing between the disciplines of applied mathematics, physics and computer science, with applications in others including social sciences, neuroscience and biology.

This book brings together leading investigators who represent various aspects of brain dynamics with the goal of presenting state-of-the-art current progress and address future developments. The individual chapters cover several fascinating facets of contemporary neuroscience from elementary computation of neurons, mesoscopic network oscillations, internally generated assembly sequences in the service of cognition, large-scale neuronal interactions within and across systems, the impact of sleep on cognition, memory, motor-sensory integration, spatial navigation, large-scale computation and consciousness. Each of these topics require appropriate levels of analyses with sufficiently high temporal and spatial resolution of neuronal activity in both local and global networks, supplemented by models and theories to explain how different levels of brain dynamics interact with each other and how the failure of such interactions results in neurologic and mental disease. While such complex questions cannot be answered exhaustively by a dozen or so chapters, this volume offers a nice synthesis of current thinking and work-in-progress on micro-, meso- and macro- dynamics of the brain.

An integrative overview of network approaches to neuroscience explores the origins of brain complexity and the link between brain structure and function. Over the last decade, the study of complex networks has expanded across diverse scientific fields. Increasingly, science is concerned with the structure, behavior, and evolution of complex systems ranging from cells to ecosystems. In Networks of the Brain, Olaf Sporns describes how the integrative nature of brain function can be illuminated from a complex network perspective. Highlighting the many emerging points of contact between neuroscience and network science, the book serves to introduce network theory to neuroscientists and neuroscience to those working on theoretical network models. Sporns emphasizes how networks connect levels of organization in the brain and how they link structure to function, offering an informal and nonmathematical treatment of the subject. Networks of the Brain provides a synthesis of the sciences of complex networks and the brain that will be an essential foundation for future research.

The study of how the brain processes time is becoming one of the most important topics in systems, cellular, computational, and cognitive neuroscience, as well as in the physiologic bases of music and language. During the last and current decade, interval timing has been intensively studied in humans and animals using increasingly sophisticated approaches. This new edition of the Neurobiology of Interval Training integrates the current knowledge of animal behavior and human cognition of the passage of time in different behavioral contexts, including the perception and production of time intervals, as well as rhythmic activities. The chapters are written by the leading experts in the fields of psychophysics, functional imaging, systems neurophysiology, and musicology. The new edition features a complete updating of the content with many new chapters. The main updates are the remarkable advances in our understanding of the neural basis of temporal processing in monkeys, rodents, and humans. The notion is that the neural clock depends on the dynamics of neural populations in the motor system, and that this general internal time representation interacts with the sensory and cognitive systems depending on the timing requirements and the behavioral contingencies of a specific task. Also, this edition delineates a clearer distinction between interval-based and beat-based timing in humans.

Boththeanalysisandgenerationoftemporalpatternsarefundamental tasks ofbiological systems. Throughout the animal kingdom, every sensory modality is designed to analyze patterns ofinformation distributed over time. Human speech and music, a visual scene in whichobjects moveorastationaryscenethatwescanwithoureyes, apatternofpressurethat changesas we moveourfingertips overan object, theelectrical field detected byafish as it swimspastobjectsinastream, orthepatternofultrasonicechoesdetectedbyabatas itflies throughacave, alldependuponspecificdistributionsofinformationovertime. Itisperhaps evenmoreobviousthatallformsofaction mustincludeatimedimension. Walking, running, talking, reaching for an object, writing, orpressing keys in aparticularorderall require the generation ofspecific patterns of muscle contractions distributed over time. Finally, most forms of behavior require a transformation from a temporal pattern of sensory input to a temporalpatternofmotoroutput, aswell as interactivemodulationofsensoryinputsystems by motoroutput systems and vice versa. Despitethefactthattheprocessesofanimallifeareinseparablefromthetimedimension, mostexperimental and theoreticalresearch on neuralcircuitry hasemphasizedtheencoding ofstatic or spatially distributed information. Within recent years, a large body ofdata has becomeavailableregardingthetimecourseoffundamentalneuralprocesses, sothatwefinally haveatourdisposalsomeofthenecessarytoolsandinformationtodiscovermechanismsused by neural circuitry to deal with time. The study of how the nervous system represents information distributed overtime is currently an exciting new frontier in neurobiology, and one in which rapid progress is likely to be made overthe next decade.

This book focuses on the theoretical side of temporal network research and gives an overview of the state of the art in the field. Curated by two pioneers in the field who have helped to shape it, the book contains contributions from many leading researchers. Temporal networks fill the border area between network science and time-series analysis and are relevant for epidemic modeling, optimization of transportation and logistics, as well as understanding biological phenomena. Over the past 20 years, network theory has proven to be one of the most powerful tools for studying and analyzing complex systems. Temporal network theory is perhaps the most recent significant development in the field in recent years, with direct applications to many of the “big data” sets. This book appeals to students, researchers, and professionals interested in theory and temporal networks—a field that has grown tremendously over the last decade. This second edition of Temporal Network Theory extends the first with three chapters highlighting recent developments in the interface with machine learning.

I know that most men, including those at ease with the problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives. Joseph Ford quoting Tolstoy (Gleick, 1987) We are used to thinking that natural objects have a certain form and that this form is determined by a characteristic scale. If we magnify the object beyond this scale, no new features are revealed. To correctly measure the properties of the object, such as length, area, or volume, we measure it at a resolution finer than the characteristic scale of the object. We expect that the value we measure has a unique value for the object. This simple idea is the basis of the calculus, Euclidean geometry, and the theory of measurement. However, Mandelbrot (1977, 1983) brought to the world's attention that many natural objects simply do not have this preconceived form. Many of the structures in space and processes in time of living things have a very different form. Living things have structures in space and fluctuations in time that cannot be characterized by one spatial or temporal scale. They extend over many spatial or temporal scales.