We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
15/2008
Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron
Henry Tuckwell
Abstract
For the Fitzhugh-Nagumo system with space-time white noise we use numerical methods to consider the generation of action potentials and the reliability of transmission in the presence of noise. The accuracy of simulated solutions is verified by comparison with known exact analytical results. Noise of small amplitude may prevent transmission directly whereas larger amplitude noise may also interfere by producing secondary non-local responses. The probability of transmission as a function of noise amplitude is found for both uniform noise and noise restricted to a patch.
For certain parameter ranges the recovery variable may be neglected to give a single component nonlinear diffusion with space-time white noise. In this case analytical results are obtained for small perturbations and noise which agree well with simulation results. For the voltage variable expressions are given for the mean, covariance and variance, and their steady state forms.
The spectral density of the voltage is also obtained. Numerical examples are given of the difference between the properties of nonlinear and linear cables and the validity of the expressions obtained for the statistical properties are investigated as a function of noise amplitude. For given parameters, analytical results are in good agreement with simulation until a certain critical noise amplitude is reached which can be estimated. The role of trigger zones in increasing the reliability of transmission is discussed.