Analysis of a FitzHugh-Nagumo-Rall model of a neuronal network

Pursuing an investigation started in (Math. Meth. Appl. Sci. 2007; 30:681-706), we consider a generalization of the FitzHugh-Nagumo model for the propagation of impulses in a network of nerve fibres. To this aim, we consider a whole neuronal network that includes models for axons, somata, dendrites, and synapses (of both inhibitory and excitatory type). We investigate separately the linear part by means of sesquilinear forms, in order to obtain well posedness and some qualitative properties. Once they are obtained, we perturb the linear problem by a nonlinear term and we prove existence of local solutions. Qualitative properties with biological meaning are also investigated.