The University of ManchesterHasan Mohammed O AlzubaidiDoctor of PhilosophyNumerical simulations of the spiking activity and the related ¯rst exittime of stochastic neural systemsSeptember 28, 2011The aim of this thesis was to study, using numerical simulation techniques, the possible effects of an additive noise on the firing properties of stochastic neural models, and the related first exit time problems. The research is divided into three main investigations. First, using SDELab, mathematical software for solving stochastic differential equations within MATLAB, we examine the influence of an additive noise on the output spike trains for the space-clamped Hodgkin Huxley (HH) model and the spatially-extended FitzHugh Nagumo (FHN) system. We find that a suitable amount of additive noise can enhance the regularity of the repetitive spiking of the space-clamped HH model. Meanwhile, we find the FHN system to be sensitive to noise, requiring that very small values of noise are chosen, in order to produce regular spikes. Second, under additive noise, we use fixed and exponential time-stepping Euler algorithms, with boundary tests, to calculate the mean first exit times (MFET) for one-dimensional neural diffusion models, represented by a stochastic space-clamped FHN system and the Ornstein-Uhlenbeck (OU) model. The strategies and theory behind these numerical methods and their convergence rates in the MFET are also considered. We find that, for different values of noise, these methods with boundary tests can improve the rate of convergence from order one half to order one, which coincides with previous studies. Finally, we look at spatially-extended systems, represented by the Barkley system with additive noise that is white in time and correlated in space, calculating mean nucleation times and mean lifetimes of traveling waves, using an efficient numerical simulation. A simple model of the dynamics of the underlying Barkley model is introduced, in order to compute the mean lifetimes, particulary for interacting waves. The reduced model is easy to use and allows us to explore the full dynamics of the kinks and antikinks, in particular over long periods. One application of the reduced model is to calculate the mean number of kinks at a given time and use this to obtain the probability that the system is excitable at a given position. With these three investigations into the effects of additive noise on stochastic neural models, we have demonstrated some of the interesting results that can be achieved using numerical techniques. We hope to extend this work, in the future, to include the effects of multiplicative noise.
|Date of Award||31 Dec 2011|
- The University of Manchester
|Supervisor||Tony Shardlow (Supervisor) & Catherine Powell (Supervisor)|