Propagation of bursting oscillations

Abstract

We investigate a system of partial differential equations of reaction-diffusion type which displays propagation of bursting oscillations. This system represents the time evolution of an assembly of cells constituted by a small nucleus of bursting cells near the origin immersed in the middle of excitable cells. We show that this system displays a global attractor in an appropriated functional space. Numerical simulations show the existence in this attractor of recurrent solutions which are waves propagating from the central source. The propagation seems possible if the excitability of the neighbouring cells is above some threshold.

Large time behaviour and synchronization of complex networks of reaction–diffusion systems of FitzHugh–Nagumo type

“…Conditions (16) and (17) generalize condition (12). They are not very restrictive and include functions Φ with spatial heterogeneity, that allow rich behavior, bifurcations and pattern formation (see [1,4]). We deduce from (16) and (17) that for all j ∈ {1, ..., d − s},…”

“…If we add boundary conditions to (1), we obtain a general reaction-diffusion system. We will not go into details concerning the existence of the semi-group of (1). We refer to [19,20,25,29] or [15,17,22,27,28], for classical results on the existence of semi-group in L p (Ω) or in C k,α (Ω) spaces.…”

Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type

“…FHN equations are a two dimensional model for oscillations and excitability. They allow to generate action potential propagation, see [1]. One can obtain FHN equations from Hodgkin-Huxley equations, by substituting a variable by its asymptotic value, using a linear correlation between two other variables and exploiting the cubic and linear shape of null-clines.…”

“…Conditions (16) and (17) generalize condition (12). They allow to choose functions Φ with spatial heterogeneity, which may lead to rich behavior, bifurcations and pattern formation (see [1]). We deduce from (17) that for all j ∈ {1, ..., d − s},…”

From biological and clinical experiments to mathematical models

“…Article 10, 'Propagation of bursting oscillations' by Ambrosio & Françoise (2009), concerns periodically slowly driven excitable dynamics and their bifurcations. The mathematical model can be seen either as a very large number of linearly coupled dynamical systems of this type or as the reaction-diffusion PDE system obtained by adding a Laplacian (dimension 2 is considered here).…”