Stochastic FitzHugh-Nagumo equations on networks with impulsive noise
Stefano Bonaccorsi, Università di Trento
Carlo Marinelli, Universität Bonn
Giacomo Ziglio, Università di Trento
Abstract
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 Lévy noise. We
prove global well-posedness in the mild sense for such stochastic
partial differential equation by monotonicity methods.
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