Preprint 60/2007

Nonlinear effects in white-noise driven spatial diffusion: general analytical results

Henry Tuckwell

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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
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

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 formula3 and the 4-point covariance to order formula5. 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.

18.10.2019, 02:13