Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

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Collective dynamics of small-world networks

I have developed "tennis elbow" from lugging this book around the past four weeks, but it is worth the pain, the effort, and the aspirin. It is also worth the (relatively speaking) bargain price. Including appendixes, this book contains 894 pages of text. The entire panorama of the neural sciences is surveyed and examined, and it is comprehensive in its scope, from genomes to social behaviors. The editors explicitly state that the book is designed as "an introductory text for students of biology, behavior, and medicine," but it is hard to imagine any audience, interested in any fragment of neuroscience at any level of sophistication, that would not enjoy this book. The editors have done a masterful job of weaving together the biologic, the behavioral, and the clinical sciences into a single tapestry in which everyone from the molecular biologist to the practicing psychiatrist can find and appreciate his or

Principles of Neural Science

http://people.physik.hu-berlin.de/~lindner/PDF/physreport.pdf

Effects of noise in excitable systems

Complex bodily rhythms are ubiquitous in living organisms. These rhythms arise from stochastic, nonlinear biological mechanisms interacting with a fluctuating environment. Disease often leads to alterations from normal to pathological rhythm. Fundamental questions concerning the dynamics of these rhythmic processes abound. For example, what is the origin of physiological rhythms? How do the rhythms interact with each other and the external environment? Can we decode the fluctuations in physiological rhythms to better diagnose human disease? And can we develop better methods to control pathological rhythms? Mathematical and physical techniques combined with physiological and medical studies are addressing these questions and are transforming our understanding of the rhythms of life.

Synchronization and rhythmic processes in physiology

What happens in our brain when we make a decision? What triggers a neuron to send out a signal? What is the neural code? This textbook for advanced undergraduate and beginning graduate students provides a thorough and up-to-date introduction to the fields of computational and theoretical neuroscience. It covers classical topics, including the Hodgkin-Huxley equations and Hopfield model, as well as modern developments in the field such as Generalized Linear Models and decision theory. Concepts are introduced using clear step-by-step explanations suitable for readers with only a basic knowledge of differential equations and probabilities, and are richly illustrated by figures and worked-out examples. End-of-chapter summaries and classroom-tested exercises make the book ideal for courses or for self-study. The authors also give pointers to the literature and an extensive bibliography, which will prove invaluable to readers interested in further study.

Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition

STOCHASTIC resonance1‡-4 (SR) is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular, non-zero level of noise5 ‡-7. SR has been proposed as a means for improving signal detection in a wide variety of systems, including superconducting quantum interference devices8, and may be used in some natural systems such as sensory neurons9‡-15. But for SR to be effective in a singleunit system (such as a sensory neuron or a single ion channel), the optimal intensity of the noise must be adjusted as the nature of the signal to be detected changes15. This has been thought to impose a limitation on the practical and natural uses of SR. Here we show that the ability of a summing network of excitable units to detect a range of weak (sub-threshold) signals (either periodic or aperiodic) can be optimized by a fixed level of noise, irrespective of the nature of the input signal. We also show that this noise does not significantly degrade the ability of the network to detect suprathreshold signals. Thus, large nonlinear networks do not suffer from the limitations of SR in single units, and might be able to use a single noise level, such as that provided by the intrinsic noise of the individual components, to enhance the system's sensitivity to weak inputs. This suggests a functional role for neuronal noise14,16‡-18 in sensory systems.

Stochastic resonance without tuning

Noise-enhanced tactile sensation

Stochastic resonance (SR) is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise. Here we present a method and theory for characterizing SR-type behavior in excitable systems with aperiodic inputs. These developments demonstrate that noise can serve to enhance the response of a nonlinear system to a weak input signal, regardless of whether the signal is periodic or aperiodic.

Aperiodic stochastic resonance in excitable systems.

1. Aperiodic stochastic resonance (ASR) is a phenomenon wherein the response of a nonlinear system to a weak aperiodic input signal is optimized by the presence of a particular, nonzero level of no...

Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance.

Stochastic resonance (SR) is a phenomenon wherein the response of a nonlinear system to a weak periodic input signal is optimized by the presence of a particular level of noise. Recently, we presented a method and theory for characterizing SR-type behavior in excitable systems with aperiodic (i.e., broadband) input signals [Phys. Rev. E 52, R3321(1995)]. We coined the term aperiodic stochastic resonance (ASR) to describe this general type of behavior. In that earlier study, we demonstrated ASR in the FitzHugh-Nagumo neuronal model. Here we demonstrate ASR in three additional systems: a bistable-well system, an integrate-and-fire neuronal model, and the Hodgkin-Huxley (HH) neuronal model. We present computational and theoretical results for each system. In the context of the HH model, we develop a general theory for ASR in excitable membranes. This work clearly shows that SR-type behavior is not limited to systems with periodic inputs. Thus, in general, noise can serve to enhance the response of a nonlinear system to a weak input signal, regardless of whether the signal is periodic or aperiodic. \textcopyright{} 1996 The American Physical Society.

Thomas T. Imhoff

Papers

Stochastic resonance without tuning

Noise-enhanced tactile sensation

Aperiodic stochastic resonance in excitable systems.

Noise-enhanced information transmission in rat SA1 cutaneous mechanoreceptors via aperiodic stochastic resonance.

Aperiodic stochastic resonance.