Full discretisation of semi-linear stochastic wave equations driven by multiplicative noise

A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretisation and thus do not suffer from a step size restriction as in the often used St\"ormer-Verlet-leap-frog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results

The effect of noise in models of spiny dendrites

The dendritic tree provides the surface area for synaptic connections between the 100 billion neurons in the brain. 90% of excitatory synapses are made onto dendritic spines which are constantly changing shape and strength. This adaptation is believed to be an important factor in learning, memory and computations within the dendritic tree. The environment in which the neuron sits is inherently noisy due to the activity in nearby neurons and the stochastic nature of synaptic gating. Therefore the effects of noise is a very important aspect in any realistic model. This work provides a comprehensive study of two spiny dendrite models driven by different forms of noise in the spine dynamics or in the membrane voltage. We investigate the effect of the noise on signal propagation along the dendrite and how any correlation in the noise may affect this behaviour. We discover a difference in the results of the two models which suggests that the form of spine connectivity is important. We also show that both models have the capacity to act as a robust filter and that a branched structure can perform logic computations

Period-doubling density waves in a chain

The authors consider a one-dimensional chain of N+2 identical particles with nearest-neighbour Lennard-Jones interaction and uniform friction. The chain is driven by a prescribed periodic motion of one end particle, with frequency v and 'strength' parameter alpha . The other end particle is held fixed. They demonstrate numerically that there is a region in the alpha -v plane where the chain has a stable state in which a density wave runs to and fro between the two ends of the chain, similarly to a ball bouncing between two walls. More importantly, they observe a period-doubling transition to chaos, for fixed v and increasing alpha , while the localised (solitary wave) character of the motion is preserve

Convergence analysis of trigonometric methods for stiff second-order stochastic differential equations

We study a class of numerical methods for a system of second-order SDE driven by a linear fast force generating high frequency oscillatory solutions. The proposed schemes permit the use of large step sizes, have uniform global error bounds in the position (i.e. independent of the large frequencies present in the SDE) and offer various additional properties. This new family of numerical integrators for SDE can be viewed as a stochastic generalisation of the trigonometric integrators for highly oscillatory deterministic problem