Search Results

Very Fast Oscillations

Roger D. Roger and Miles A. Whittington

in Cortical Oscillations in Health and Disease

VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast ... More

VFO occurs in in vitro models when chemical receptors are blocked. In particular, VFO does not require GABAA receptors, even though interneurons fire at high rates during in vivo very fast oscillations. VFO can be accounted for by a model in which neuronal spiking percolates through a sparse network of electrically coupled axons. This model predicts that VFO frequency depends on gap junction conductance, mediated by an effect on crossing time (i.e. the time it takes for a spike in one axon to elicit a spike in a coupled axon, estimated to be of order 0.2 ms). VFO in cerebellar slices also depends on gap junctions, but the physical principles are slightly different: cerebellar VFO appears to depend on many:one propagation of spiking, in effect a form of axonal coincidence detection.Less

Percolation and epidemics

Sergey N. Dorogovtsev

in Lectures on Complex Networks

This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to ... More

This chapter considers the organization of connected components in uncorrelated networks, in particular, the structure and size of a giant connected component. These properties are closely related to the percolation properties of these networks. The value of the percolation threshold for various degree distributions is estimated, and the resilience of scale-free networks against random failures is described. The hierarchical organization of k-cores in these networks is discussed. A few basic epidemic models on complex networks are introduced, and the evolution of diseases in networks is described.Less

The Pleasures and Pains of Studying the Two-Type Richardson Model

Maria Deijfen and Olle Häggström

in Analysis and Stochastics of Growth Processes and Interface Models

This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the ... More

This chapter provides a survey of known results and open problems for the two-type Richardson model, which is a stochastic model for competition on a lattice. In its simplest formulation, the Richardson model describes the evolution of a single infectious entity on the lattice, but more recently the dynamics have been extended to comprise two competing growing entities. For this version of the model, the main question is whether there is a positive probability for both entities to simultaneously grow to occupy infinite parts of the lattice, the conjecture being that the answer is yes if and only if the entities have the same intensity. In this paper attention focuses on the two-type model, but the most important results for the one-type version are also described.Less

Applications of the Lace Expansion to Statistical-Mechanical Models

Akira Sakai

in Analysis and Stochastics of Growth Processes and Interface Models

Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical ... More

Synergetics is a common feature in interesting statistical-mechanical problems. One of the most important examples of synergetics is the emergence of a second-order phase transition and critical behaviour. It is rich and still far from fully understood. The reason why it is so difficult is due to the increase to infinity of the number of strongly correlated variables in the vicinity of the critical point. For example, the Ising model, which is a model for magnets, exhibits critical behaviour as the temperature comes closer to its critical value; the closer the temperature is to criticality, the more spin variables cooperate with each other to attain the global magnetization. In this regime, neither standard probability theory for independent random variables nor naive perturbation techniques work. The lace expansion, which is the topic of this article, is currently one of the few approaches to rigorous investigation of critical behaviour for various statistical-mechanical models. The chapter summarizes some of the most intriguing lace-expansion results for self-avoiding walk (SAW), percolation, and the Ising model.Less

PERCOLATIVE INGREDIENTS

Mathew Penrose

in Random Geometric Graphs

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) ... More

This chapter contains some known results on connectivity which are used later on. The notion of unicoherence of a simply-connected set is explained and extended to lattices. Peierls (counting) arguments are described for estimating the number of connected sets in the lattice, and elements of (lattice) percolation theory are described. A multiparameter ergodic theorem is given, and the basic theory of continuum percolation is described. Some of the theory of Poisson point processes are recalled, including the superposition, thinning, and scaling theorems.Less

Information Percolation in OTC Markets

Darrell Duffie

in Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets

This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the ... More

This chapter describes a simple model of the “percolation” of information of common interest through an over-the-counter market with many agents. It also includes an explicit solution for the cross-sectional distribution of posterior beliefs at each time. It begins with the basic information structure for the economy and the setting for search and random matching. It then shows how to solve the model for the dynamics of the cross-sectional distribution of information. The remainder of the chapter is devoted to market settings and to extensions of the model that handle public releases of information, the receipt of new private information over time, and the release of information among groups of more than two agents at a time.Less

Growth Phenomena in Cellular Automata

Janko Gravner

in New Constructions in Cellular Automata

We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that ... More

We illustrate growth phenomena in two-dimensional cellular automata (CA) by four case studies. The first CA, which we call Obstacle Course, describes the effect that obstacles have on such features of simple growth models as linear expansion and coherent asymptotic shape. Our next CA is random-walk-based Internal Diffusion Limited Aggregation, which spreads sublinearly, but with a shape which can be explicitly computed due to hydrodynamic effects. Then we propose a simple scheme for characterizing CA according to their growth properties, as indicated by two Larger than Life examples. Finally, a very simple case of Spatial Prisoner’s Dilemma illustrates nucleation analysis of CA. In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e.g., Meakin [34] and more than 1300 references cited in the latter). Cellular automata (CA) growth models, by virtue of their simplicity and amenability to computer experimentation [25], have become particularly popular in the last 20 years, especially in physics research literature [40, 42]. Needless to say, precise mathematical results are hard to come by, and many basic questions remain completely open at the rigorous level. The purpose of this chapter, then, is to outline some successes of the mathematical approach and to identify some fundamental difficulties. We will mainly address three themes which can be summarized by the terms: aggregation, nucleation, and constraint-expansion transition. These themes also provide opportunities to touch on the roles of randomness, monotonicity, and linearity in CA investigations. We choose to illustrate these issues by particular CA rules, with little attempt to formulate a general theory. Simplicity is often, and rightly, touted as an important selling point of cellular automata. We have, therefore, tried to choose the simplest models which, while being amenable to some mathematical analysis, raise a host of intriguing unanswered questions. The next few paragraphs outline subsequent sections of this chapter. Aggregation models typically study properties of growth from a small initial seed. Arguably, the simplest dynamics are obtained by adding sites on the boundary in a uniform fashion. Less

Stochastic Geometry and Telecommunications Networks

Sergei Zuyev

in New Perspectives in Stochastic Geometry

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and ... More

Just as queueing theory revolutionized the study of circuit switched telephony in the twentieth century, stochastic geometry is gradually becoming a necessary theoretical tool for modelling and analysis of modern telecommunications systems, in which spatial arrangement is typically a crucial consideration in their performance evaluation, optimization or future development. In this survey we aim to summarize the main stochastic geometry models and tools currently used in studying modern telecommunications. We outline specifics of wired, wireless fixed and ad hoc systems and show how stochastic geometry modelling helps in their analysis and optimization. Point and line processes, Palm theory, shot‐noise processes, random tessellations, Boolean models, percolation, random graphs and networks, spatial statistics and optimization: this is a far from exhaustive list of techniques used in studying contemporary telecommunications systems and which we shall briefly discuss.Less

Percolation and Random Graphs

Remco van der Hofstad

in New Perspectives in Stochastic Geometry

In this chapter, we define percolation and random graph models, and survey the features of these models.

Applications to Other Wh-Fronting Languages, Pied-Piping, and Intervention Effects

Seth Cable

in The Grammar of Q: Q-Particles, Wh-Movement, and Pied-Piping

This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These ... More

This chapter begins the argument that the ‘Q-based’ account should be extended to all other wh-fronting languages. It first presents some general typological and learning-theoretic arguments. These include (i) the fact that the ‘Q-based’ account is transparently motivated for other wh-fronting languages (such as Edo), and (ii) the ability for the ‘Q-based’ account to provide a unified theory of the ill-formedness of P-stranding and left-branch extractions across languages. Following this, the consequences of the account for the theory of pied-piping structures are examined. In particular, it is shown that extending the ‘Q-based’ account to all other wh-fronting languages entails that the phenomenon dubbed ‘pied-piping’ need not exist at all. Finally, the chapter develops a ‘Q-based’ theory of multiple wh-questions in English and German. It is shown that the analysis predicts the complementary distribution of Superiority Effects and Intervention Effects in these languages. Finally, Intervention Effects in pied-piping structures are examined, and the ‘Q-based’ theory is shown to make an accurate (and surprising) prediction.Less

Optimization

Sergey N. Dorogovtsev

in Lectures on Complex Networks

This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and ... More

This chapter demonstrates how a universal requirement for optimality leads to the complex structural organization of a network. It discusses a long-lasting criticism of the preferential concept and describes existing approaches to the optimization-driven evolution of complex networks. In particular, the optimized trade-off model of a growing network is described, as well as models showing the explosive percolation phenomenon.Less

Percolation and network resilience

Mark Newman

in Networks

A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are ... More

A discussion of the site percolation process on networks and its application as a model of network resilience. The chapter starts with a description of the percolation process, in which nodes are randomly removed from a network, and of the percolation phase transition at which a giant percolating cluster forms. The properties of percolation on configuration model networks are studied, including networks with power-law degree distributions, and including both uniform and non-uniform removal of nodes. Computer algorithms for simulating percolation on real-world networks are also discussed, and numerical results are given for several example networks, including the internet and a social network.Less

ELECTRON MOBILITY IN DENSE HE GAS

A.F. Borghesani

in Ions and electrons in liquid helium

Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility ... More

Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility clearly shows that the formation of electron bubbles is a continuous phenomenon. Localization of electrons in bubbles also appears at high temperatures if the density is so large that the free energy of the localized state is negative enough. Percolation and hydrodynamic models have been devised to explain the continuous transition from high-mobility states to low-mobility states. It is shown that density-dependent, quantum multiple scattering effects modify the energy of the nearly free electron in a way that can be accurately described by heuristically modifying the kinetic theory prediction.Less

HOPPING CONDUCTIVITY

V.F. Gantmakher

in Electrons and Disorder in Solids

This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition ... More

This chapter is dedicated to electron transitions between localized states. After a description of the states with the help of localization radius and the general expression for the transition probability between two states — the hopping conductivity theory — is presented. The theory is based on Abrahams-Miller network of random resistances modelling an insulator and the percolation theory. Three types of hopping conductivity are presented: nearest-neighbour hopping, variable-range hopping with Coulomb gap at the Fermi level (Efros-Shklovskii law), and variable-range hopping without Coulomb gap (Mott law). They can be distinguished by the temperature dependence of the activated conductance.Less