

Preprint 60/2007
Nonlinear effects in white-noise driven spatial diffusion: general analytical results
Henry Tuckwell
Contact the author: Please use for correspondence this email.
Submission date: 28. Jun. 2007
published in: Physica / A, 387 (2008) 7, p. 1455-1463
DOI number (of the published article): 10.1016/j.physa.2007.10.062
Bibtex
with the following different title: Nonlinear effects in white-noise driven spatial diffusion : general analytical results and probabilities of of exceeding threshold
Keywords and phrases: neurobiology, stochastic, PDE
Abstract:
We consider a general nonlinear diffusion, typified by those
deriving from Fitzhugh-Nagumo or Hindmarsh-Rose models of nerve-cell dynamics,
perturbed also by 2-parameter white noise. In order to investigate the effects
of the nonlinearity, we
find for general boundary conditions the mean to order and the 4-point covariance to order
.
The derivations involve multiple stochastic integrals in the plane. The mean and variance of the
state variable are thus obtained and may be used to estimate the probabilities that
a threshold value is exceeded as a function of space and time. An example is
reported on results for white noise driven diffusion with a cubic nonlinearity.
From the asymptotic form of the covariance the spectral density of the process can also
be obtained.