Eric Foxall

A scaling law for random walks on networks

The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

A Contraction Argument for Two-Dimensional Spiking Neuron Models

A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs.

Scaling properties of paths on graphs

Let

Social contact processes and the partner model

G

Tragedy of the commons in the chemostat

be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on

Survival and extinction results for a patch model with sexual reproduction

G

Critical behaviour of the partner model

. Fix vertices

Explicit construction of chaotic attractors in Glass networks

u

The Naming Game on the complete graph

and

Stochastic calculus and sample path estimation for jump processes

v