Stochastic FitzHugh-Nagumo equations on networks with impulsive noise

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.

Optimal control of stochastic FitzHugh–Nagumo equation

“…In Sec. 2 we shall prove the well-posedness of problem (1.1), see [9] for other results of this type.…”

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

“…It shall be noticed that in [3], more general, possibly nonlocal, forms of the matrix B are considered.…”

“…Let us briefly introduce our model, following the approach of [3,4]. Let G = (V, E) be a graph with n vertices and m edges; to each node v i ∈ V is associated a variable p i (t) evolving according to the stochastic equation ∂ ∂t p i (t) = −b i (t)p i (t) + k i (t) + σ i (t, p i (t)) ∂ ∂t L(t, v i ), (1.1) for all t ∈ R and i = 1, .…”

“…In case of constant coefficients, the dynamics of the linearized system is described by an evolution semigroup e t A on a suitable product space H; properties of this semigroup are given in [3,4]. In case of time-varying coefficients, the same system is described by a family of evolution operators U(t, s) generated by the non-autonomous perturbation A(t) = A + V(t).…”

Stochastic reaction-diffusion equations on networks with dynamic time-delayed boundary conditions

“…In fact, there is a broad area of possible applications where the mathematical use of graphs and random dynamics stated on them, play a crucial role, as in the case, e.g., of quantum mechanics, see, e.g. [36], the books [24,31] and references therein; in neurobiology, as an example concerning the study of stochastic system of the FitzHugh-Nagumo type, see, e.g., [1,2,3,8,10]; or in finance, see, e.g., [6,14,15,26] and references therein, particularly in the light of numerical applications, see, e.g., [18] Concerning the aforementioned ambit, a possible approach which has shown to be particularly useful, is to introduce a suitable infinite dimensional space of functions that takes into account the underlying graph domain and then tackle the diffusion problem exploiting both functional analytic tools and infinite dimensional analysis. This technique had led to a systematic study of Stochastic Partial Differential Equations (SPDEs) on networks, showing that it is in general possible to rewrite a diffusion problem defined on a network in a general abstract form, see, e.g., [8,10,11,19], and the monograph [33] for a detailed introduction to the subject.…”