The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

Neurons in the brain communicate by short electrical pulses, the so-called action potentials or spikes. How can we understand the process of spike generation? How can we understand information transmission by neurons? What happens if thousands of neurons are coupled together in a seemingly random network? How does the network connectivity determine the activity patterns? And, vice versa, how does the spike activity influence the connectivity pattern? These questions are addressed in this 2002 introduction to spiking neurons aimed at those taking courses in computational neuroscience, theoretical biology, biophysics, or neural networks. The approach will suit students of physics, mathematics, or computer science; it will also be useful for biologists who are interested in mathematical modelling. The text is enhanced by many worked examples and illustrations. There are no mathematical prerequisites beyond what the audience would meet as undergraduates: more advanced techniques are introduced in an elementary, concrete fashion when needed.

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Spiking Neuron Models: Single Neurons, Populations, Plasticity

Noise in dynamical systems is usually considered a nuisance. But in certain nonlinear systems, including electronic circuits and biological sensory apparatus, the presence of noise can in fact enhance the detection of weak signals. This phenomenon, called stochastic resonance, may find useful application in physical, technological and biomedical contexts.

Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs

A broad body of experimental work has demonstrated that apparently spontaneous brain activity is not random. At the level of large-scale neural systems, as measured with functional MRI (fMRI), this ongoing activity reflects the organization of a series of highly coherent functional networks. These so-called resting-state networks (RSNs) closely relate to the underlying anatomical connectivity but cannot be understood in those terms alone. Here we review three large-scale neural system models of primate neocortex that emphasize the key contributions of local dynamics, signal transmission delays and noise to the emerging RSNs. We propose that the formation and dissolution of resting-state patterns reflects the exploration of possible functional network configurations around a stable anatomical skeleton.

Emerging concepts for the dynamical organization of resting-state activity in the brain

The concept of self-organized criticality was introduced to explain the behaviour of the sandpile model. In this model, particles are randomly dropped onto a square grid of boxes. When a box accumulates four particles they are redistributed to the four adjacent boxes or lost off the edge of the grid. Redistributions can lead to further instabilities with the possibility of more particles being lost from the grid, contributing to the size of each ‘avalanche’. These model ‘avalanches’ satisfied a power-law frequency‐area distribution with a slope near unity. Other cellular-automata models, including the slider-block and forest-fire models, are also said to exhibit self-organized critical behaviour. It has been argued that earthquakes, landslides, forest fires, and species extinctions are examples of self-organized criticality in nature. In addition, wars and stock market crashes have been associated with this behaviour. The forest-fire model is particularly interesting in terms of its relation to the critical-point behaviour of the sitepercolation model. In the basic forest-fire model, trees are randomly planted on a grid of points. Periodically in time, sparks are randomly dropped on the grid. If a spark drops on a tree, that tree and adjacent trees burn in a model fire. The fires are the ‘avalanches’ and they are found to satisfy power-law frequency‐area distributions with slopes near unity. This forest-fire model is closely related to the site-percolation model, that exhibits critical behaviour. In the forest-fire model there is an inverse cascade of trees from small clusters to large clusters, trees are lost primarily from model fires that destroy the largest clusters. This quasi steady-state cascade gives a power-law frequency‐area distribution for both clusters of trees and smaller fires. The site-percolation model is equivalent to the forest-fire model without fires. In this case there is a transient cascade of trees from small to large clusters and a power-law distribution is found only at a critical density of trees.

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Self-organized criticality

The concept of stochastic resonance has been introduced previously to describe a curious phenomenon in bistable systems subject to both periodic and random forcing: an increase in the input noise can result in an improvement in the output signal-to-noise ratio. In this paper we present a detailed theoretical and numerical study of stochastic resonance, based on a rate equation approach. The main result is an equation for the output signal-to-noise ratio as a function of the rate at which noise induces hopping between the two states. The manner in which the input noise strength determines this hopping rate depends on the precise nature of the bistable system. For this reason, the theory is applied to two classes of bistable systems, the double-well (continuous) system and the two-state (discrete) system. The theory is tested in detail against digital simulations.

Theory of stochastic resonance.

We report the first observation of stochastic resonance in an optical device, the bidirectional ring laser. The experiment exploits a new technique to modulate periodically the asymmetry between the two counter-rotating lasing modes. The measurements verify that the addition of injected noise can lead to an improved signal-to-noise ratio (relative to that observed with no externally injected noise).

Observation of stochastic resonance in a ring laser.

Near the onset of a dynamical instability, any time-periodic system can act to amplify small periodic perturbations. The details of this small-signal sensitivity depend solely on the type of bifurcation involved: Explicit expressions are derived for the power spectra in the vicinity of the simplest classes of codimension-1 bifurcations. Results obtained from analog simulations of a period-doubling system are in good agreement with the theory. We propose that the superconducting Josephson-junction parametric amplifier is an example of this amplification process.

Small-signal amplification in bifurcating dynamical systems

Near the onset of a period-doubling bifurcation, any dynamical system can be used to amplify perturbations near half the fundamental frequency: The closer the bifurcation point, the greater the amplification. An analytic expression for the frequency response curve is derived explicitly for the driven Duffing oscillator. Results of analog simulations are presented to check the main features of the theory. We propose that the superconducting Josephson parametric amplifier is an example of this amplification process.

Period-doubling systems as small-signal amplifiers.

We study a deterministic version of the cellular automaton first shown to display self-organized criticality. Detailed analysis shows that there exist many coexisting periodic attractors, with a period that is independent of initial condition. This leads us to picture the critical state as the union of many such coexisting, neutrally stable orbits. Dhar's recently developed formalism [Phys. Rev. Lett. 64, 1613 (1990)] can be used to explain many of the observed regularities.

Bruce McNamara

Papers

Theory of stochastic resonance.

Observation of stochastic resonance in a ring laser.

Small-signal amplification in bifurcating dynamical systems

Period-doubling systems as small-signal amplifiers.

Self-organized criticality in a deterministic automaton.