Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

Details Stochastic FitzHugh-Nagumo equations on networks with impulsive noise Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

2007-12-04

arxiv.org/abs/0712.0580v2

Mathematics

Analysis of PDEs

18 pages. Minor revision

Scientific paper

We consider a system of nonlinear partial differential equations with stochastic dynamical boundary conditions that arises in models of neurophysiology for the diffusion of electrical potentials through a finite network of neurons. Motivated by the discussion in the biological literature, we impose a general diffusion equation on each edge through a generalized version of the FitzHugh-Nagumo model, while the noise acting on the boundary is described by a generalized stochastic Kirchhoff law on the nodes. In the abstract framework of matrix operators theory, we rewrite this stochastic boundary value problem as a stochastic evolution equation in infinite dimensions with a power-type nonlinearity, driven by an additive L\'evy noise. We prove global well-posedness in the mild sense for such stochastic partial differential equation by monotonicity methods.

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