Abstract:
Part I
In this first talk, I will introduce a toy nonlinear model which can be seen as 
a toy model for the FitzHugh-Nagumo (FHN) Reaction-Diffusion (RD) Model 
well known in Mathematical Neuroscience. I will provide a comprehensive 
analysis of the linearized system as well as global convergence results 
toward particular solutions for the nonlinear system.  The talk will end with
numerical illustrations of solutions.

Part II
In this second  talk, I will replace the toy model of part I in its context and 
point out arising general questions: I will connect the toy model with the 
FHN model. I will recall the origin of FHN and its relevance in Neuroscience 
context. I will provide  theoretical results obtained recently for 
nonhomogeneous FHN and exhibit numerical solutions. I will conclude by 
listing some questions linked with the two talks.