Stochastic Fitzhugh-nagumo Equations on Networks with Impulsive Noise

In this paper we study a system of nonlinear diffusion equations on a finite network in the presence of an impulsive noise acting on the nodes of the system. We allow a rather general nonlinear drift term, including dissipative functions of FitzHugh-Nagumo type (i.e. f(u) = −u(u− 1)(u− a)) arising in various models of neurophysiology (see e.g. the monograph [19] for more details). Electric signaling by neurons has been studied since the 50s, starting with the now classical Hodgkin-Huxley model [16] for the diffusion of the transmembrane electrical potential in a neuronal cell. This model consists of a system of four equations describing the diffusion of the electrical potential and the behavior of various ion channels. Successive simplifications of the model, trying to capture the key phenomena of the Hodgkin-Huxley model, lead to the reduced FitzHugh-Nagumo equation, which is a scalar equation with two stable states (see e.g. [27]). Among other papers dealing with the case of a whole neuronal network (usually modeled as a graph with m edges and n nodes), which is intended to be a simplified model for a large region of the brain, let us mention a series of recent papers by Mugnolo et al. [21, 25], where the well-posedness of the isolated system is studied. Note that, for a diffusion on a network, other conditions must be imposed in order to define the behavior at the nodes. We impose a continuity condition, that is, given any node in the network, the electrical potentials of all its incident edges are equal. Each edge represents an active soma, and in this part of the cell the potential evolves following a generalized Kirchhoff condition that we model with dynamical boundary conditions for the internal dynamics. Since the classical work of Walsh [28], stochastic partial differential equations have been an important modeling tools in neurophysiology, where a random forcing is introduced to model several external perturbations acting on the system. In our neural network, we model the electrical activity of