Nonlinear effects in white-noise driven spatial diffusion: General analytical results and probabilities of exceeding threshold

Henry C. Tuckwell

Physica A: Statistical Mechanics and its Applications, 2008, vol. 387, issue 7, 1455-1463

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 ϵ2 and the four-point covariance to order ϵ3. 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. A numerical example is given for a space–time 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.

Keywords: Nonlinear diffusion; Stochastic partial differential equations; Neurons (search for similar items in EconPapers)
Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:387:y:2008:i:7:p:1455-1463

DOI: 10.1016/j.physa.2007.10.062

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